How do you find #lim sin(2x)/ln(x+1)# as #x->0# using l'Hospital's Rule?

When one of these indeterminate forms is present, l'Hopital's rule states that:

So, we can take the numerator and denominator separately, as per l'Hopital's rule:

To find the limit ( \lim_{x \to 0} \frac{\sin(2x)}{\ln(x+1)} ) using L'Hôpital's Rule, we can differentiate the numerator and denominator separately with respect to ( x ) until we get a determinate form (0/0 or ∞/∞), and then evaluate the limit.

1. Differentiate the numerator and denominator: [ \lim_{x \to 0} \frac{\sin(2x)}{\ln(x+1)} ] [ = \lim_{x \to 0} \frac{d}{dx}\left(\frac{\sin(2x)}{\ln(x+1)}\right) ] [ = \lim_{x \to 0} \frac{2\cos(2x)}{\frac{1}{x+1}} ]

2. Simplify the expression: [ = \To find the limit ( \lim_{x \to 0} \frac{\sin(2x)}{\ln(x+1)} ) using L'Hôpital's Rule, we can differentiate the numerator and denominator separately with respect to ( x ) until we get a determinate form (0/0 or ∞/∞), and then evaluate the limit.

3. Differentiate the numerator and denominator: [ \lim_{x \to 0} \frac{\sin(2x)}{\ln(x+1)} ] [ = \lim_{x \to 0} \frac{d}{dx}\left(\frac{\sin(2x)}{\ln(x+1)}\right) ] [ = \lim_{x \to 0} \frac{2\cos(2x)}{\frac{1}{x+1}} ]

4. Simplify the expression: [ = \limTo find the limit ( \lim_{x \to 0} \frac{\sin(2x)}{\ln(x+1)} ) using L'Hôpital's Rule, we can differentiate the numerator and denominator separately with respect to ( x ) until we get a determinate form (0/0 or ∞/∞), and then evaluate the limit.

5. Differentiate the numerator and denominator: [ \lim_{x \to 0} \frac{\sin(2x)}{\ln(x+1)} ] [ = \lim_{x \to 0} \frac{d}{dx}\left(\frac{\sin(2x)}{\ln(x+1)}\right) ] [ = \lim_{x \to 0} \frac{2\cos(2x)}{\frac{1}{x+1}} ]

6. Simplify the expression: [ = \lim_{To find the limit ( \lim_{x \to 0} \frac{\sin(2x)}{\ln(x+1)} ) using L'Hôpital's Rule, we can differentiate the numerator and denominator separately with respect to ( x ) until we get a determinate form (0/0 or ∞/∞), and then evaluate the limit.

7. Differentiate the numerator and denominator: [ \lim_{x \to 0} \frac{\sin(2x)}{\ln(x+1)} ] [ = \lim_{x \to 0} \frac{d}{dx}\left(\frac{\sin(2x)}{\ln(x+1)}\right) ] [ = \lim_{x \to 0} \frac{2\cos(2x)}{\frac{1}{x+1}} ]

8. Simplify the expression: [ = \lim_{xTo find the limit ( \lim_{x \to 0} \frac{\sin(2x)}{\ln(x+1)} ) using L'Hôpital's Rule, we can differentiate the numerator and denominator separately with respect to ( x ) until we get a determinate form (0/0 or ∞/∞), and then evaluate the limit.

9. Differentiate the numerator and denominator: [ \lim_{x \to 0} \frac{\sin(2x)}{\ln(x+1)} ] [ = \lim_{x \to 0} \frac{d}{dx}\left(\frac{\sin(2x)}{\ln(x+1)}\right) ] [ = \lim_{x \to 0} \frac{2\cos(2x)}{\frac{1}{x+1}} ]

10. Simplify the expression: [ = \lim_{x \To find the limit ( \lim_{x \to 0} \frac{\sin(2x)}{\ln(x+1)} ) using L'Hôpital's Rule, we can differentiate the numerator and denominator separately with respect to ( x ) until we get a determinate form (0/0 or ∞/∞), and then evaluate the limit.

11. Differentiate the numerator and denominator: [ \lim_{x \to 0} \frac{\sin(2x)}{\ln(x+1)} ] [ = \lim_{x \to 0} \frac{d}{dx}\left(\frac{\sin(2x)}{\ln(x+1)}\right) ] [ = \lim_{x \to 0} \frac{2\cos(2x)}{\frac{1}{x+1}} ]

12. Simplify the expression: [ = \lim_{x \toTo find the limit ( \lim_{x \to 0} \frac{\sin(2x)}{\ln(x+1)} ) using L'Hôpital's Rule, we need to differentiate the numeratorTo find the limit ( \lim_{x \to 0} \frac{\sin(2x)}{\ln(x+1)} ) using L'Hôpital's Rule, we can differentiate the numerator and denominator separately with respect to ( x ) until we get a determinate form (0/0 or ∞/∞), and then evaluate the limit.

13. Differentiate the numerator and denominator: [ \lim_{x \to 0} \frac{\sin(2x)}{\ln(x+1)} ] [ = \lim_{x \to 0} \frac{d}{dx}\left(\frac{\sin(2x)}{\ln(x+1)}\right) ] [ = \lim_{x \to 0} \frac{2\cos(2x)}{\frac{1}{x+1}} ]

14. Simplify the expression: [ = \lim_{x \to To find the limit ( \lim_{x \to 0} \frac{\sin(2x)}{\ln(x+1)} ) using L'Hôpital's Rule, we need to differentiate the numerator and the denominator separately with respect to ( x ) until we reach a determinate form.

To find the limit ( \lim_{x \to 0} \frac{\sin(2x)}{\ln(x+1)} ) using L'Hôpital's Rule, we can differentiate the numerator and denominator separately with respect to ( x ) until we get a determinate form (0/0 or ∞/∞), and then evaluate the limit.

1. Differentiate the numerator and denominator: [ \lim_{x \to 0} \frac{\sin(2x)}{\ln(x+1)} ] [ = \lim_{x \to 0} \frac{d}{dx}\left(\frac{\sin(2x)}{\ln(x+1)}\right) ] [ = \lim_{x \to 0} \frac{2\cos(2x)}{\frac{1}{x+1}} ]

2. Simplify the expression: [ = \lim_{x \to 0To find the limit ( \lim_{x \to 0} \frac{\sin(2x)}{\ln(x+1)} ) using L'Hôpital's Rule, we need to differentiate the numerator and the denominator separately with respect to ( x ) until we reach a determinate form.

3. Differentiate the numerator: ( \frac{d}{dx} \sin(2x) = 2\To find the limit ( \lim_{x \to 0} \frac{\sin(2x)}{\ln(x+1)} ) using L'Hôpital's Rule, we can differentiate the numerator and denominator separately with respect to ( x ) until we get a determinate form (0/0 or ∞/∞), and then evaluate the limit.

4. Differentiate the numerator and denominator: [ \lim_{x \to 0} \frac{\sin(2x)}{\ln(x+1)} ] [ = \lim_{x \to 0} \frac{d}{dx}\left(\frac{\sin(2x)}{\ln(x+1)}\right) ] [ = \lim_{x \to 0} \frac{2\cos(2x)}{\frac{1}{x+1}} ]

5. Simplify the expression: [ = \lim_{x \to 0}To find the limit ( \lim_{x \to 0} \frac{\sin(2x)}{\ln(x+1)} ) using L'Hôpital's Rule, we need to differentiate the numerator and the denominator separately with respect to ( x ) until we reach a determinate form.

6. Differentiate the numerator: ( \frac{d}{dx} \sin(2x) = 2\cos(2x) ).

7. Differentiate the denominator: ( \frac{d}{dx} \ln(xTo find the limit ( \lim_{x \to 0} \frac{\sin(2x)}{\ln(x+1)} ) using L'Hôpital's Rule, we can differentiate the numerator and denominator separately with respect to ( x ) until we get a determinate form (0/0 or ∞/∞), and then evaluate the limit.

8. Differentiate the numerator and denominator: [ \lim_{x \to 0} \frac{\sin(2x)}{\ln(x+1)} ] [ = \lim_{x \to 0} \frac{d}{dx}\left(\frac{\sin(2x)}{\ln(x+1)}\right) ] [ = \lim_{x \to 0} \frac{2\cos(2x)}{\frac{1}{x+1}} ]

9. Simplify the expression: [ = \lim_{x \to 0} To find the limit ( \lim_{x \to 0} \frac{\sin(2x)}{\ln(x+1)} ) using L'Hôpital's Rule, we need to differentiate the numerator and the denominator separately with respect to ( x ) until we reach a determinate form.

10. Differentiate the numerator: ( \frac{d}{dx} \sin(2x) = 2\cos(2x) ).

11. Differentiate the denominator: ( \frac{d}{dx} \ln(x+1) = \frac{1}{x+1} ).

Now, we evaluate theTo find the limit ( \lim_{x \to 0} \frac{\sin(2x)}{\ln(x+1)} ) using L'Hôpital's Rule, we can differentiate the numerator and denominator separately with respect to ( x ) until we get a determinate form (0/0 or ∞/∞), and then evaluate the limit.

1. Differentiate the numerator and denominator: [ \lim_{x \to 0} \frac{\sin(2x)}{\ln(x+1)} ] [ = \lim_{x \to 0} \frac{d}{dx}\left(\frac{\sin(2x)}{\ln(x+1)}\right) ] [ = \lim_{x \to 0} \frac{2\cos(2x)}{\frac{1}{x+1}} ]

2. Simplify the expression: [ = \lim_{x \to 0} 2To find the limit ( \lim_{x \to 0} \frac{\sin(2x)}{\ln(x+1)} ) using L'Hôpital's Rule, we need to differentiate the numerator and the denominator separately with respect to ( x ) until we reach a determinate form.

3. Differentiate the numerator: ( \frac{d}{dx} \sin(2x) = 2\cos(2x) ).

4. Differentiate the denominator: ( \frac{d}{dx} \ln(x+1) = \frac{1}{x+1} ).

Now, we evaluate the limit of the ratio of the derivatives:

[To find the limit ( \lim_{x \to 0} \frac{\sin(2x)}{\ln(x+1)} ) using L'Hôpital's Rule, we can differentiate the numerator and denominator separately with respect to ( x ) until we get a determinate form (0/0 or ∞/∞), and then evaluate the limit.

1. Differentiate the numerator and denominator: [ \lim_{x \to 0} \frac{\sin(2x)}{\ln(x+1)} ] [ = \lim_{x \to 0} \frac{d}{dx}\left(\frac{\sin(2x)}{\ln(x+1)}\right) ] [ = \lim_{x \to 0} \frac{2\cos(2x)}{\frac{1}{x+1}} ]

2. Simplify the expression: [ = \lim_{x \to 0} 2\To find the limit ( \lim_{x \to 0} \frac{\sin(2x)}{\ln(x+1)} ) using L'Hôpital's Rule, we need to differentiate the numerator and the denominator separately with respect to ( x ) until we reach a determinate form.

3. Differentiate the numerator: ( \frac{d}{dx} \sin(2x) = 2\cos(2x) ).

4. Differentiate the denominator: ( \frac{d}{dx} \ln(x+1) = \frac{1}{x+1} ).

Now, we evaluate the limit of the ratio of the derivatives:

[ \lim_{x \to 0} \frac{To find the limit ( \lim_{x \to 0} \frac{\sin(2x)}{\ln(x+1)} ) using L'Hôpital's Rule, we can differentiate the numerator and denominator separately with respect to ( x ) until we get a determinate form (0/0 or ∞/∞), and then evaluate the limit.

1. Differentiate the numerator and denominator: [ \lim_{x \to 0} \frac{\sin(2x)}{\ln(x+1)} ] [ = \lim_{x \to 0} \frac{d}{dx}\left(\frac{\sin(2x)}{\ln(x+1)}\right) ] [ = \lim_{x \to 0} \frac{2\cos(2x)}{\frac{1}{x+1}} ]

2. Simplify the expression: [ = \lim_{x \to 0} 2\cosTo find the limit ( \lim_{x \to 0} \frac{\sin(2x)}{\ln(x+1)} ) using L'Hôpital's Rule, we need to differentiate the numerator and the denominator separately with respect to ( x ) until we reach a determinate form.

3. Differentiate the numerator: ( \frac{d}{dx} \sin(2x) = 2\cos(2x) ).

4. Differentiate the denominator: ( \frac{d}{dx} \ln(x+1) = \frac{1}{x+1} ).

Now, we evaluate the limit of the ratio of the derivatives:

[ \lim_{x \to 0} \frac{2\cos(2xTo find the limit ( \lim_{x \to 0} \frac{\sin(2x)}{\ln(x+1)} ) using L'Hôpital's Rule, we can differentiate the numerator and denominator separately with respect to ( x ) until we get a determinate form (0/0 or ∞/∞), and then evaluate the limit.

1. Differentiate the numerator and denominator: [ \lim_{x \to 0} \frac{\sin(2x)}{\ln(x+1)} ] [ = \lim_{x \to 0} \frac{d}{dx}\left(\frac{\sin(2x)}{\ln(x+1)}\right) ] [ = \lim_{x \to 0} \frac{2\cos(2x)}{\frac{1}{x+1}} ]

2. Simplify the expression: [ = \lim_{x \to 0} 2\cos(To find the limit ( \lim_{x \to 0} \frac{\sin(2x)}{\ln(x+1)} ) using L'Hôpital's Rule, we need to differentiate the numerator and the denominator separately with respect to ( x ) until we reach a determinate form.

3. Differentiate the numerator: ( \frac{d}{dx} \sin(2x) = 2\cos(2x) ).

4. Differentiate the denominator: ( \frac{d}{dx} \ln(x+1) = \frac{1}{x+1} ).

Now, we evaluate the limit of the ratio of the derivatives:

[ \lim_{x \to 0} \frac{2\cos(2x)}{\frac{1}{x+1}} \To find the limit ( \lim_{x \to 0} \frac{\sin(2x)}{\ln(x+1)} ) using L'Hôpital's Rule, we can differentiate the numerator and denominator separately with respect to ( x ) until we get a determinate form (0/0 or ∞/∞), and then evaluate the limit.

1. Differentiate the numerator and denominator: [ \lim_{x \to 0} \frac{\sin(2x)}{\ln(x+1)} ] [ = \lim_{x \to 0} \frac{d}{dx}\left(\frac{\sin(2x)}{\ln(x+1)}\right) ] [ = \lim_{x \to 0} \frac{2\cos(2x)}{\frac{1}{x+1}} ]

2. Simplify the expression: [ = \lim_{x \to 0} 2\cos(2To find the limit ( \lim_{x \to 0} \frac{\sin(2x)}{\ln(x+1)} ) using L'Hôpital's Rule, we need to differentiate the numerator and the denominator separately with respect to ( x ) until we reach a determinate form.

3. Differentiate the numerator: ( \frac{d}{dx} \sin(2x) = 2\cos(2x) ).

4. Differentiate the denominator: ( \frac{d}{dx} \ln(x+1) = \frac{1}{x+1} ).

Now, we evaluate the limit of the ratio of the derivatives:

[ \lim_{x \to 0} \frac{2\cos(2x)}{\frac{1}{x+1}} ]

At ( x = 0 ),To find the limit ( \lim_{x \to 0} \frac{\sin(2x)}{\ln(x+1)} ) using L'Hôpital's Rule, we can differentiate the numerator and denominator separately with respect to ( x ) until we get a determinate form (0/0 or ∞/∞), and then evaluate the limit.

1. Differentiate the numerator and denominator: [ \lim_{x \to 0} \frac{\sin(2x)}{\ln(x+1)} ] [ = \lim_{x \to 0} \frac{d}{dx}\left(\frac{\sin(2x)}{\ln(x+1)}\right) ] [ = \lim_{x \to 0} \frac{2\cos(2x)}{\frac{1}{x+1}} ]

2. Simplify the expression: [ = \lim_{x \to 0} 2\cos(2xTo find the limit ( \lim_{x \to 0} \frac{\sin(2x)}{\ln(x+1)} ) using L'Hôpital's Rule, we need to differentiate the numerator and the denominator separately with respect to ( x ) until we reach a determinate form.

3. Differentiate the numerator: ( \frac{d}{dx} \sin(2x) = 2\cos(2x) ).

4. Differentiate the denominator: ( \frac{d}{dx} \ln(x+1) = \frac{1}{x+1} ).

Now, we evaluate the limit of the ratio of the derivatives:

[ \lim_{x \to 0} \frac{2\cos(2x)}{\frac{1}{x+1}} ]

At ( x = 0 ), the denominator becomes ( \frac{1To find the limit ( \lim_{x \to 0} \frac{\sin(2x)}{\ln(x+1)} ) using L'Hôpital's Rule, we can differentiate the numerator and denominator separately with respect to ( x ) until we get a determinate form (0/0 or ∞/∞), and then evaluate the limit.

1. Differentiate the numerator and denominator: [ \lim_{x \to 0} \frac{\sin(2x)}{\ln(x+1)} ] [ = \lim_{x \to 0} \frac{d}{dx}\left(\frac{\sin(2x)}{\ln(x+1)}\right) ] [ = \lim_{x \to 0} \frac{2\cos(2x)}{\frac{1}{x+1}} ]

2. Simplify the expression: [ = \lim_{x \to 0} 2\cos(2x)(To find the limit ( \lim_{x \to 0} \frac{\sin(2x)}{\ln(x+1)} ) using L'Hôpital's Rule, we need to differentiate the numerator and the denominator separately with respect to ( x ) until we reach a determinate form.

3. Differentiate the numerator: ( \frac{d}{dx} \sin(2x) = 2\cos(2x) ).

4. Differentiate the denominator: ( \frac{d}{dx} \ln(x+1) = \frac{1}{x+1} ).

Now, we evaluate the limit of the ratio of the derivatives:

[ \lim_{x \to 0} \frac{2\cos(2x)}{\frac{1}{x+1}} ]

At ( x = 0 ), the denominator becomes ( \frac{1}{1} = 1 ) and the numeratorTo find the limit ( \lim_{x \to 0} \frac{\sin(2x)}{\ln(x+1)} ) using L'Hôpital's Rule, we can differentiate the numerator and denominator separately with respect to ( x ) until we get a determinate form (0/0 or ∞/∞), and then evaluate the limit.

1. Differentiate the numerator and denominator: [ \lim_{x \to 0} \frac{\sin(2x)}{\ln(x+1)} ] [ = \lim_{x \to 0} \frac{d}{dx}\left(\frac{\sin(2x)}{\ln(x+1)}\right) ] [ = \lim_{x \to 0} \frac{2\cos(2x)}{\frac{1}{x+1}} ]

2. Simplify the expression: [ = \lim_{x \to 0} 2\cos(2x)(x+To find the limit ( \lim_{x \to 0} \frac{\sin(2x)}{\ln(x+1)} ) using L'Hôpital's Rule, we need to differentiate the numerator and the denominator separately with respect to ( x ) until we reach a determinate form.

3. Differentiate the numerator: ( \frac{d}{dx} \sin(2x) = 2\cos(2x) ).

4. Differentiate the denominator: ( \frac{d}{dx} \ln(x+1) = \frac{1}{x+1} ).

Now, we evaluate the limit of the ratio of the derivatives:

[ \lim_{x \to 0} \frac{2\cos(2x)}{\frac{1}{x+1}} ]

At ( x = 0 ), the denominator becomes ( \frac{1}{1} = 1 ) and the numerator becomesTo find the limit ( \lim_{x \to 0} \frac{\sin(2x)}{\ln(x+1)} ) using L'Hôpital's Rule, we can differentiate the numerator and denominator separately with respect to ( x ) until we get a determinate form (0/0 or ∞/∞), and then evaluate the limit.

1. Differentiate the numerator and denominator: [ \lim_{x \to 0} \frac{\sin(2x)}{\ln(x+1)} ] [ = \lim_{x \to 0} \frac{d}{dx}\left(\frac{\sin(2x)}{\ln(x+1)}\right) ] [ = \lim_{x \to 0} \frac{2\cos(2x)}{\frac{1}{x+1}} ]

2. Simplify the expression: [ = \lim_{x \to 0} 2\cos(2x)(x+1To find the limit ( \lim_{x \to 0} \frac{\sin(2x)}{\ln(x+1)} ) using L'Hôpital's Rule, we need to differentiate the numerator and the denominator separately with respect to ( x ) until we reach a determinate form.

3. Differentiate the numerator: ( \frac{d}{dx} \sin(2x) = 2\cos(2x) ).

4. Differentiate the denominator: ( \frac{d}{dx} \ln(x+1) = \frac{1}{x+1} ).

Now, we evaluate the limit of the ratio of the derivatives:

[ \lim_{x \to 0} \frac{2\cos(2x)}{\frac{1}{x+1}} ]

At ( x = 0 ), the denominator becomes ( \frac{1}{1} = 1 ) and the numerator becomes (To find the limit ( \lim_{x \to 0} \frac{\sin(2x)}{\ln(x+1)} ) using L'Hôpital's Rule, we can differentiate the numerator and denominator separately with respect to ( x ) until we get a determinate form (0/0 or ∞/∞), and then evaluate the limit.

1. Differentiate the numerator and denominator: [ \lim_{x \to 0} \frac{\sin(2x)}{\ln(x+1)} ] [ = \lim_{x \to 0} \frac{d}{dx}\left(\frac{\sin(2x)}{\ln(x+1)}\right) ] [ = \lim_{x \to 0} \frac{2\cos(2x)}{\frac{1}{x+1}} ]

2. Simplify the expression: [ = \lim_{x \to 0} 2\cos(2x)(x+1) \To find the limit ( \lim_{x \to 0} \frac{\sin(2x)}{\ln(x+1)} ) using L'Hôpital's Rule, we need to differentiate the numerator and the denominator separately with respect to ( x ) until we reach a determinate form.

3. Differentiate the numerator: ( \frac{d}{dx} \sin(2x) = 2\cos(2x) ).

4. Differentiate the denominator: ( \frac{d}{dx} \ln(x+1) = \frac{1}{x+1} ).

Now, we evaluate the limit of the ratio of the derivatives:

[ \lim_{x \to 0} \frac{2\cos(2x)}{\frac{1}{x+1}} ]

At ( x = 0 ), the denominator becomes ( \frac{1}{1} = 1 ) and the numerator becomes ( 2To find the limit ( \lim_{x \to 0} \frac{\sin(2x)}{\ln(x+1)} ) using L'Hôpital's Rule, we can differentiate the numerator and denominator separately with respect to ( x ) until we get a determinate form (0/0 or ∞/∞), and then evaluate the limit.

1. Differentiate the numerator and denominator: [ \lim_{x \to 0} \frac{\sin(2x)}{\ln(x+1)} ] [ = \lim_{x \to 0} \frac{d}{dx}\left(\frac{\sin(2x)}{\ln(x+1)}\right) ] [ = \lim_{x \to 0} \frac{2\cos(2x)}{\frac{1}{x+1}} ]

2. Simplify the expression: [ = \lim_{x \to 0} 2\cos(2x)(x+1) ]

To find the limit ( \lim_{x \to 0} \frac{\sin(2x)}{\ln(x+1)} ) using L'Hôpital's Rule, we need to differentiate the numerator and the denominator separately with respect to ( x ) until we reach a determinate form.

1. Differentiate the numerator: ( \frac{d}{dx} \sin(2x) = 2\cos(2x) ).
2. Differentiate the denominator: ( \frac{d}{dx} \ln(x+1) = \frac{1}{x+1} ).

Now, we evaluate the limit of the ratio of the derivatives:

[ \lim_{x \to 0} \frac{2\cos(2x)}{\frac{1}{x+1}} ]

At ( x = 0 ), the denominator becomes ( \frac{1}{1} = 1 ) and the numerator becomes ( 2\To find the limit ( \lim_{x \to 0} \frac{\sin(2x)}{\ln(x+1)} ) using L'Hôpital's Rule, we can differentiate the numerator and denominator separately with respect to ( x ) until we get a determinate form (0/0 or ∞/∞), and then evaluate the limit.

1. Differentiate the numerator and denominator: [ \lim_{x \to 0} \frac{\sin(2x)}{\ln(x+1)} ] [ = \lim_{x \to 0} \frac{d}{dx}\left(\frac{\sin(2x)}{\ln(x+1)}\right) ] [ = \lim_{x \to 0} \frac{2\cos(2x)}{\frac{1}{x+1}} ]

2. Simplify the expression: [ = \lim_{x \to 0} 2\cos(2x)(x+1) ]

3To find the limit ( \lim_{x \to 0} \frac{\sin(2x)}{\ln(x+1)} ) using L'Hôpital's Rule, we need to differentiate the numerator and the denominator separately with respect to ( x ) until we reach a determinate form.

1. Differentiate the numerator: ( \frac{d}{dx} \sin(2x) = 2\cos(2x) ).
2. Differentiate the denominator: ( \frac{d}{dx} \ln(x+1) = \frac{1}{x+1} ).

Now, we evaluate the limit of the ratio of the derivatives:

[ \lim_{x \to 0} \frac{2\cos(2x)}{\frac{1}{x+1}} ]

At ( x = 0 ), the denominator becomes ( \frac{1}{1} = 1 ) and the numerator becomes ( 2\cosTo find the limit ( \lim_{x \to 0} \frac{\sin(2x)}{\ln(x+1)} ) using L'Hôpital's Rule, we can differentiate the numerator and denominator separately with respect to ( x ) until we get a determinate form (0/0 or ∞/∞), and then evaluate the limit.

1. Differentiate the numerator and denominator: [ \lim_{x \to 0} \frac{\sin(2x)}{\ln(x+1)} ] [ = \lim_{x \to 0} \frac{d}{dx}\left(\frac{\sin(2x)}{\ln(x+1)}\right) ] [ = \lim_{x \to 0} \frac{2\cos(2x)}{\frac{1}{x+1}} ]

2. Simplify the expression: [ = \lim_{x \to 0} 2\cos(2x)(x+1) ]

3.To find the limit ( \lim_{x \to 0} \frac{\sin(2x)}{\ln(x+1)} ) using L'Hôpital's Rule, we need to differentiate the numerator and the denominator separately with respect to ( x ) until we reach a determinate form.

1. Differentiate the numerator: ( \frac{d}{dx} \sin(2x) = 2\cos(2x) ).
2. Differentiate the denominator: ( \frac{d}{dx} \ln(x+1) = \frac{1}{x+1} ).

Now, we evaluate the limit of the ratio of the derivatives:

[ \lim_{x \to 0} \frac{2\cos(2x)}{\frac{1}{x+1}} ]

At ( x = 0 ), the denominator becomes ( \frac{1}{1} = 1 ) and the numerator becomes ( 2\cos(0To find the limit ( \lim_{x \to 0} \frac{\sin(2x)}{\ln(x+1)} ) using L'Hôpital's Rule, we can differentiate the numerator and denominator separately with respect to ( x ) until we get a determinate form (0/0 or ∞/∞), and then evaluate the limit.

1. Differentiate the numerator and denominator: [ \lim_{x \to 0} \frac{\sin(2x)}{\ln(x+1)} ] [ = \lim_{x \to 0} \frac{d}{dx}\left(\frac{\sin(2x)}{\ln(x+1)}\right) ] [ = \lim_{x \to 0} \frac{2\cos(2x)}{\frac{1}{x+1}} ]

2. Simplify the expression: [ = \lim_{x \to 0} 2\cos(2x)(x+1) ]

3. EvaluateTo find the limit ( \lim_{x \to 0} \frac{\sin(2x)}{\ln(x+1)} ) using L'Hôpital's Rule, we need to differentiate the numerator and the denominator separately with respect to ( x ) until we reach a determinate form.

4. Differentiate the numerator: ( \frac{d}{dx} \sin(2x) = 2\cos(2x) ).

5. Differentiate the denominator: ( \frac{d}{dx} \ln(x+1) = \frac{1}{x+1} ).

Now, we evaluate the limit of the ratio of the derivatives:

[ \lim_{x \to 0} \frac{2\cos(2x)}{\frac{1}{x+1}} ]

At ( x = 0 ), the denominator becomes ( \frac{1}{1} = 1 ) and the numerator becomes ( 2\cos(0) =To find the limit ( \lim_{x \to 0} \frac{\sin(2x)}{\ln(x+1)} ) using L'Hôpital's Rule, we can differentiate the numerator and denominator separately with respect to ( x ) until we get a determinate form (0/0 or ∞/∞), and then evaluate the limit.

1. Differentiate the numerator and denominator: [ \lim_{x \to 0} \frac{\sin(2x)}{\ln(x+1)} ] [ = \lim_{x \to 0} \frac{d}{dx}\left(\frac{\sin(2x)}{\ln(x+1)}\right) ] [ = \lim_{x \to 0} \frac{2\cos(2x)}{\frac{1}{x+1}} ]

2. Simplify the expression: [ = \lim_{x \to 0} 2\cos(2x)(x+1) ]

3. Evaluate theTo find the limit ( \lim_{x \to 0} \frac{\sin(2x)}{\ln(x+1)} ) using L'Hôpital's Rule, we need to differentiate the numerator and the denominator separately with respect to ( x ) until we reach a determinate form.

4. Differentiate the numerator: ( \frac{d}{dx} \sin(2x) = 2\cos(2x) ).

5. Differentiate the denominator: ( \frac{d}{dx} \ln(x+1) = \frac{1}{x+1} ).

Now, we evaluate the limit of the ratio of the derivatives:

[ \lim_{x \to 0} \frac{2\cos(2x)}{\frac{1}{x+1}} ]

At ( x = 0 ), the denominator becomes ( \frac{1}{1} = 1 ) and the numerator becomes ( 2\cos(0) = To find the limit ( \lim_{x \to 0} \frac{\sin(2x)}{\ln(x+1)} ) using L'Hôpital's Rule, we can differentiate the numerator and denominator separately with respect to ( x ) until we get a determinate form (0/0 or ∞/∞), and then evaluate the limit.

1. Differentiate the numerator and denominator: [ \lim_{x \to 0} \frac{\sin(2x)}{\ln(x+1)} ] [ = \lim_{x \to 0} \frac{d}{dx}\left(\frac{\sin(2x)}{\ln(x+1)}\right) ] [ = \lim_{x \to 0} \frac{2\cos(2x)}{\frac{1}{x+1}} ]

2. Simplify the expression: [ = \lim_{x \to 0} 2\cos(2x)(x+1) ]

3. Evaluate the limitTo find the limit ( \lim_{x \to 0} \frac{\sin(2x)}{\ln(x+1)} ) using L'Hôpital's Rule, we need to differentiate the numerator and the denominator separately with respect to ( x ) until we reach a determinate form.

4. Differentiate the numerator: ( \frac{d}{dx} \sin(2x) = 2\cos(2x) ).

5. Differentiate the denominator: ( \frac{d}{dx} \ln(x+1) = \frac{1}{x+1} ).

Now, we evaluate the limit of the ratio of the derivatives:

[ \lim_{x \to 0} \frac{2\cos(2x)}{\frac{1}{x+1}} ]

At ( x = 0 ), the denominator becomes ( \frac{1}{1} = 1 ) and the numerator becomes ( 2\cos(0) = 2To find the limit ( \lim_{x \to 0} \frac{\sin(2x)}{\ln(x+1)} ) using L'Hôpital's Rule, we can differentiate the numerator and denominator separately with respect to ( x ) until we get a determinate form (0/0 or ∞/∞), and then evaluate the limit.

1. Differentiate the numerator and denominator: [ \lim_{x \to 0} \frac{\sin(2x)}{\ln(x+1)} ] [ = \lim_{x \to 0} \frac{d}{dx}\left(\frac{\sin(2x)}{\ln(x+1)}\right) ] [ = \lim_{x \to 0} \frac{2\cos(2x)}{\frac{1}{x+1}} ]

2. Simplify the expression: [ = \lim_{x \to 0} 2\cos(2x)(x+1) ]

3. Evaluate the limit asTo find the limit ( \lim_{x \to 0} \frac{\sin(2x)}{\ln(x+1)} ) using L'Hôpital's Rule, we need to differentiate the numerator and the denominator separately with respect to ( x ) until we reach a determinate form.

4. Differentiate the numerator: ( \frac{d}{dx} \sin(2x) = 2\cos(2x) ).

5. Differentiate the denominator: ( \frac{d}{dx} \ln(x+1) = \frac{1}{x+1} ).

Now, we evaluate the limit of the ratio of the derivatives:

[ \lim_{x \to 0} \frac{2\cos(2x)}{\frac{1}{x+1}} ]

At ( x = 0 ), the denominator becomes ( \frac{1}{1} = 1 ) and the numerator becomes ( 2\cos(0) = 2 \To find the limit ( \lim_{x \to 0} \frac{\sin(2x)}{\ln(x+1)} ) using L'Hôpital's Rule, we can differentiate the numerator and denominator separately with respect to ( x ) until we get a determinate form (0/0 or ∞/∞), and then evaluate the limit.

1. Differentiate the numerator and denominator: [ \lim_{x \to 0} \frac{\sin(2x)}{\ln(x+1)} ] [ = \lim_{x \to 0} \frac{d}{dx}\left(\frac{\sin(2x)}{\ln(x+1)}\right) ] [ = \lim_{x \to 0} \frac{2\cos(2x)}{\frac{1}{x+1}} ]

2. Simplify the expression: [ = \lim_{x \to 0} 2\cos(2x)(x+1) ]

3. Evaluate the limit as (To find the limit ( \lim_{x \to 0} \frac{\sin(2x)}{\ln(x+1)} ) using L'Hôpital's Rule, we need to differentiate the numerator and the denominator separately with respect to ( x ) until we reach a determinate form.

4. Differentiate the numerator: ( \frac{d}{dx} \sin(2x) = 2\cos(2x) ).

5. Differentiate the denominator: ( \frac{d}{dx} \ln(x+1) = \frac{1}{x+1} ).

Now, we evaluate the limit of the ratio of the derivatives:

[ \lim_{x \to 0} \frac{2\cos(2x)}{\frac{1}{x+1}} ]

At ( x = 0 ), the denominator becomes ( \frac{1}{1} = 1 ) and the numerator becomes ( 2\cos(0) = 2 ).

To find the limit ( \lim_{x \to 0} \frac{\sin(2x)}{\ln(x+1)} ) using L'Hôpital's Rule, we can differentiate the numerator and denominator separately with respect to ( x ) until we get a determinate form (0/0 or ∞/∞), and then evaluate the limit.

1. Differentiate the numerator and denominator: [ \lim_{x \to 0} \frac{\sin(2x)}{\ln(x+1)} ] [ = \lim_{x \to 0} \frac{d}{dx}\left(\frac{\sin(2x)}{\ln(x+1)}\right) ] [ = \lim_{x \to 0} \frac{2\cos(2x)}{\frac{1}{x+1}} ]

2. Simplify the expression: [ = \lim_{x \to 0} 2\cos(2x)(x+1) ]

3. Evaluate the limit as ( xTo find the limit ( \lim_{x \to 0} \frac{\sin(2x)}{\ln(x+1)} ) using L'Hôpital's Rule, we need to differentiate the numerator and the denominator separately with respect to ( x ) until we reach a determinate form.

4. Differentiate the numerator: ( \frac{d}{dx} \sin(2x) = 2\cos(2x) ).

5. Differentiate the denominator: ( \frac{d}{dx} \ln(x+1) = \frac{1}{x+1} ).

Now, we evaluate the limit of the ratio of the derivatives:

[ \lim_{x \to 0} \frac{2\cos(2x)}{\frac{1}{x+1}} ]

At ( x = 0 ), the denominator becomes ( \frac{1}{1} = 1 ) and the numerator becomes ( 2\cos(0) = 2 ).

SoTo find the limit ( \lim_{x \to 0} \frac{\sin(2x)}{\ln(x+1)} ) using L'Hôpital's Rule, we can differentiate the numerator and denominator separately with respect to ( x ) until we get a determinate form (0/0 or ∞/∞), and then evaluate the limit.

1. Differentiate the numerator and denominator: [ \lim_{x \to 0} \frac{\sin(2x)}{\ln(x+1)} ] [ = \lim_{x \to 0} \frac{d}{dx}\left(\frac{\sin(2x)}{\ln(x+1)}\right) ] [ = \lim_{x \to 0} \frac{2\cos(2x)}{\frac{1}{x+1}} ]

2. Simplify the expression: [ = \lim_{x \to 0} 2\cos(2x)(x+1) ]

3. Evaluate the limit as ( x \To find the limit ( \lim_{x \to 0} \frac{\sin(2x)}{\ln(x+1)} ) using L'Hôpital's Rule, we need to differentiate the numerator and the denominator separately with respect to ( x ) until we reach a determinate form.

4. Differentiate the numerator: ( \frac{d}{dx} \sin(2x) = 2\cos(2x) ).

5. Differentiate the denominator: ( \frac{d}{dx} \ln(x+1) = \frac{1}{x+1} ).

Now, we evaluate the limit of the ratio of the derivatives:

[ \lim_{x \to 0} \frac{2\cos(2x)}{\frac{1}{x+1}} ]

At ( x = 0 ), the denominator becomes ( \frac{1}{1} = 1 ) and the numerator becomes ( 2\cos(0) = 2 ).

So,To find the limit ( \lim_{x \to 0} \frac{\sin(2x)}{\ln(x+1)} ) using L'Hôpital's Rule, we can differentiate the numerator and denominator separately with respect to ( x ) until we get a determinate form (0/0 or ∞/∞), and then evaluate the limit.

1. Differentiate the numerator and denominator: [ \lim_{x \to 0} \frac{\sin(2x)}{\ln(x+1)} ] [ = \lim_{x \to 0} \frac{d}{dx}\left(\frac{\sin(2x)}{\ln(x+1)}\right) ] [ = \lim_{x \to 0} \frac{2\cos(2x)}{\frac{1}{x+1}} ]

2. Simplify the expression: [ = \lim_{x \to 0} 2\cos(2x)(x+1) ]

3. Evaluate the limit as ( x \toTo find the limit ( \lim_{x \to 0} \frac{\sin(2x)}{\ln(x+1)} ) using L'Hôpital's Rule, we need to differentiate the numerator and the denominator separately with respect to ( x ) until we reach a determinate form.

4. Differentiate the numerator: ( \frac{d}{dx} \sin(2x) = 2\cos(2x) ).

5. Differentiate the denominator: ( \frac{d}{dx} \ln(x+1) = \frac{1}{x+1} ).

Now, we evaluate the limit of the ratio of the derivatives:

[ \lim_{x \to 0} \frac{2\cos(2x)}{\frac{1}{x+1}} ]

At ( x = 0 ), the denominator becomes ( \frac{1}{1} = 1 ) and the numerator becomes ( 2\cos(0) = 2 ).

So, the limitTo find the limit ( \lim_{x \to 0} \frac{\sin(2x)}{\ln(x+1)} ) using L'Hôpital's Rule, we can differentiate the numerator and denominator separately with respect to ( x ) until we get a determinate form (0/0 or ∞/∞), and then evaluate the limit.

1. Differentiate the numerator and denominator: [ \lim_{x \to 0} \frac{\sin(2x)}{\ln(x+1)} ] [ = \lim_{x \to 0} \frac{d}{dx}\left(\frac{\sin(2x)}{\ln(x+1)}\right) ] [ = \lim_{x \to 0} \frac{2\cos(2x)}{\frac{1}{x+1}} ]

2. Simplify the expression: [ = \lim_{x \to 0} 2\cos(2x)(x+1) ]

3. Evaluate the limit as ( x \to 0To find the limit ( \lim_{x \to 0} \frac{\sin(2x)}{\ln(x+1)} ) using L'Hôpital's Rule, we need to differentiate the numerator and the denominator separately with respect to ( x ) until we reach a determinate form.

4. Differentiate the numerator: ( \frac{d}{dx} \sin(2x) = 2\cos(2x) ).

5. Differentiate the denominator: ( \frac{d}{dx} \ln(x+1) = \frac{1}{x+1} ).

Now, we evaluate the limit of the ratio of the derivatives:

[ \lim_{x \to 0} \frac{2\cos(2x)}{\frac{1}{x+1}} ]

At ( x = 0 ), the denominator becomes ( \frac{1}{1} = 1 ) and the numerator becomes ( 2\cos(0) = 2 ).

So, the limit simplifies toTo find the limit ( \lim_{x \to 0} \frac{\sin(2x)}{\ln(x+1)} ) using L'Hôpital's Rule, we can differentiate the numerator and denominator separately with respect to ( x ) until we get a determinate form (0/0 or ∞/∞), and then evaluate the limit.

1. Differentiate the numerator and denominator: [ \lim_{x \to 0} \frac{\sin(2x)}{\ln(x+1)} ] [ = \lim_{x \to 0} \frac{d}{dx}\left(\frac{\sin(2x)}{\ln(x+1)}\right) ] [ = \lim_{x \to 0} \frac{2\cos(2x)}{\frac{1}{x+1}} ]

2. Simplify the expression: [ = \lim_{x \to 0} 2\cos(2x)(x+1) ]

3. Evaluate the limit as ( x \to 0 \To find the limit ( \lim_{x \to 0} \frac{\sin(2x)}{\ln(x+1)} ) using L'Hôpital's Rule, we need to differentiate the numerator and the denominator separately with respect to ( x ) until we reach a determinate form.

4. Differentiate the numerator: ( \frac{d}{dx} \sin(2x) = 2\cos(2x) ).

5. Differentiate the denominator: ( \frac{d}{dx} \ln(x+1) = \frac{1}{x+1} ).

Now, we evaluate the limit of the ratio of the derivatives:

[ \lim_{x \to 0} \frac{2\cos(2x)}{\frac{1}{x+1}} ]

At ( x = 0 ), the denominator becomes ( \frac{1}{1} = 1 ) and the numerator becomes ( 2\cos(0) = 2 ).

So, the limit simplifies to:

To find the limit ( \lim_{x \to 0} \frac{\sin(2x)}{\ln(x+1)} ) using L'Hôpital's Rule, we can differentiate the numerator and denominator separately with respect to ( x ) until we get a determinate form (0/0 or ∞/∞), and then evaluate the limit.

1. Differentiate the numerator and denominator: [ \lim_{x \to 0} \frac{\sin(2x)}{\ln(x+1)} ] [ = \lim_{x \to 0} \frac{d}{dx}\left(\frac{\sin(2x)}{\ln(x+1)}\right) ] [ = \lim_{x \to 0} \frac{2\cos(2x)}{\frac{1}{x+1}} ]

2. Simplify the expression: [ = \lim_{x \to 0} 2\cos(2x)(x+1) ]

3. Evaluate the limit as ( x \to 0 ): To find the limit ( \lim_{x \to 0} \frac{\sin(2x)}{\ln(x+1)} ) using L'Hôpital's Rule, we need to differentiate the numerator and the denominator separately with respect to ( x ) until we reach a determinate form.

4. Differentiate the numerator: ( \frac{d}{dx} \sin(2x) = 2\cos(2x) ).

5. Differentiate the denominator: ( \frac{d}{dx} \ln(x+1) = \frac{1}{x+1} ).

Now, we evaluate the limit of the ratio of the derivatives:

[ \lim_{x \to 0} \frac{2\cos(2x)}{\frac{1}{x+1}} ]

At ( x = 0 ), the denominator becomes ( \frac{1}{1} = 1 ) and the numerator becomes ( 2\cos(0) = 2 ).

So, the limit simplifies to:

[ \limTo find the limit ( \lim_{x \to 0} \frac{\sin(2x)}{\ln(x+1)} ) using L'Hôpital's Rule, we can differentiate the numerator and denominator separately with respect to ( x ) until we get a determinate form (0/0 or ∞/∞), and then evaluate the limit.

1. Differentiate the numerator and denominator: [ \lim_{x \to 0} \frac{\sin(2x)}{\ln(x+1)} ] [ = \lim_{x \to 0} \frac{d}{dx}\left(\frac{\sin(2x)}{\ln(x+1)}\right) ] [ = \lim_{x \to 0} \frac{2\cos(2x)}{\frac{1}{x+1}} ]

2. Simplify the expression: [ = \lim_{x \to 0} 2\cos(2x)(x+1) ]

3. Evaluate the limit as ( x \to 0 ): [To find the limit ( \lim_{x \to 0} \frac{\sin(2x)}{\ln(x+1)} ) using L'Hôpital's Rule, we need to differentiate the numerator and the denominator separately with respect to ( x ) until we reach a determinate form.

4. Differentiate the numerator: ( \frac{d}{dx} \sin(2x) = 2\cos(2x) ).

5. Differentiate the denominator: ( \frac{d}{dx} \ln(x+1) = \frac{1}{x+1} ).

Now, we evaluate the limit of the ratio of the derivatives:

[ \lim_{x \to 0} \frac{2\cos(2x)}{\frac{1}{x+1}} ]

At ( x = 0 ), the denominator becomes ( \frac{1}{1} = 1 ) and the numerator becomes ( 2\cos(0) = 2 ).

So, the limit simplifies to:

[ \lim_{x \To find the limit ( \lim_{x \to 0} \frac{\sin(2x)}{\ln(x+1)} ) using L'Hôpital's Rule, we can differentiate the numerator and denominator separately with respect to ( x ) until we get a determinate form (0/0 or ∞/∞), and then evaluate the limit.

1. Differentiate the numerator and denominator: [ \lim_{x \to 0} \frac{\sin(2x)}{\ln(x+1)} ] [ = \lim_{x \to 0} \frac{d}{dx}\left(\frac{\sin(2x)}{\ln(x+1)}\right) ] [ = \lim_{x \to 0} \frac{2\cos(2x)}{\frac{1}{x+1}} ]

2. Simplify the expression: [ = \lim_{x \to 0} 2\cos(2x)(x+1) ]

3. Evaluate the limit as ( x \to 0 ): [ =To find the limit ( \lim_{x \to 0} \frac{\sin(2x)}{\ln(x+1)} ) using L'Hôpital's Rule, we need to differentiate the numerator and the denominator separately with respect to ( x ) until we reach a determinate form.

4. Differentiate the numerator: ( \frac{d}{dx} \sin(2x) = 2\cos(2x) ).

5. Differentiate the denominator: ( \frac{d}{dx} \ln(x+1) = \frac{1}{x+1} ).

Now, we evaluate the limit of the ratio of the derivatives:

[ \lim_{x \to 0} \frac{2\cos(2x)}{\frac{1}{x+1}} ]

At ( x = 0 ), the denominator becomes ( \frac{1}{1} = 1 ) and the numerator becomes ( 2\cos(0) = 2 ).

So, the limit simplifies to:

[ \lim_{x \toTo find the limit ( \lim_{x \to 0} \frac{\sin(2x)}{\ln(x+1)} ) using L'Hôpital's Rule, we can differentiate the numerator and denominator separately with respect to ( x ) until we get a determinate form (0/0 or ∞/∞), and then evaluate the limit.

1. Differentiate the numerator and denominator: [ \lim_{x \to 0} \frac{\sin(2x)}{\ln(x+1)} ] [ = \lim_{x \to 0} \frac{d}{dx}\left(\frac{\sin(2x)}{\ln(x+1)}\right) ] [ = \lim_{x \to 0} \frac{2\cos(2x)}{\frac{1}{x+1}} ]

2. Simplify the expression: [ = \lim_{x \to 0} 2\cos(2x)(x+1) ]

3. Evaluate the limit as ( x \to 0 ): [ = To find the limit ( \lim_{x \to 0} \frac{\sin(2x)}{\ln(x+1)} ) using L'Hôpital's Rule, we need to differentiate the numerator and the denominator separately with respect to ( x ) until we reach a determinate form.

4. Differentiate the numerator: ( \frac{d}{dx} \sin(2x) = 2\cos(2x) ).

5. Differentiate the denominator: ( \frac{d}{dx} \ln(x+1) = \frac{1}{x+1} ).

Now, we evaluate the limit of the ratio of the derivatives:

[ \lim_{x \to 0} \frac{2\cos(2x)}{\frac{1}{x+1}} ]

At ( x = 0 ), the denominator becomes ( \frac{1}{1} = 1 ) and the numerator becomes ( 2\cos(0) = 2 ).

So, the limit simplifies to:

[ \lim_{x \to To find the limit ( \lim_{x \to 0} \frac{\sin(2x)}{\ln(x+1)} ) using L'Hôpital's Rule, we can differentiate the numerator and denominator separately with respect to ( x ) until we get a determinate form (0/0 or ∞/∞), and then evaluate the limit.

1. Differentiate the numerator and denominator: [ \lim_{x \to 0} \frac{\sin(2x)}{\ln(x+1)} ] [ = \lim_{x \to 0} \frac{d}{dx}\left(\frac{\sin(2x)}{\ln(x+1)}\right) ] [ = \lim_{x \to 0} \frac{2\cos(2x)}{\frac{1}{x+1}} ]

2. Simplify the expression: [ = \lim_{x \to 0} 2\cos(2x)(x+1) ]

3. Evaluate the limit as ( x \to 0 ): [ = 2To find the limit ( \lim_{x \to 0} \frac{\sin(2x)}{\ln(x+1)} ) using L'Hôpital's Rule, we need to differentiate the numerator and the denominator separately with respect to ( x ) until we reach a determinate form.

4. Differentiate the numerator: ( \frac{d}{dx} \sin(2x) = 2\cos(2x) ).

5. Differentiate the denominator: ( \frac{d}{dx} \ln(x+1) = \frac{1}{x+1} ).

Now, we evaluate the limit of the ratio of the derivatives:

[ \lim_{x \to 0} \frac{2\cos(2x)}{\frac{1}{x+1}} ]

At ( x = 0 ), the denominator becomes ( \frac{1}{1} = 1 ) and the numerator becomes ( 2\cos(0) = 2 ).

So, the limit simplifies to:

[ \lim_{x \to 0To find the limit ( \lim_{x \to 0} \frac{\sin(2x)}{\ln(x+1)} ) using L'Hôpital's Rule, we can differentiate the numerator and denominator separately with respect to ( x ) until we get a determinate form (0/0 or ∞/∞), and then evaluate the limit.

1. Differentiate the numerator and denominator: [ \lim_{x \to 0} \frac{\sin(2x)}{\ln(x+1)} ] [ = \lim_{x \to 0} \frac{d}{dx}\left(\frac{\sin(2x)}{\ln(x+1)}\right) ] [ = \lim_{x \to 0} \frac{2\cos(2x)}{\frac{1}{x+1}} ]

2. Simplify the expression: [ = \lim_{x \to 0} 2\cos(2x)(x+1) ]

3. Evaluate the limit as ( x \to 0 ): [ = 2\To find the limit ( \lim_{x \to 0} \frac{\sin(2x)}{\ln(x+1)} ) using L'Hôpital's Rule, we need to differentiate the numerator and the denominator separately with respect to ( x ) until we reach a determinate form.

4. Differentiate the numerator: ( \frac{d}{dx} \sin(2x) = 2\cos(2x) ).

5. Differentiate the denominator: ( \frac{d}{dx} \ln(x+1) = \frac{1}{x+1} ).

Now, we evaluate the limit of the ratio of the derivatives:

[ \lim_{x \to 0} \frac{2\cos(2x)}{\frac{1}{x+1}} ]

At ( x = 0 ), the denominator becomes ( \frac{1}{1} = 1 ) and the numerator becomes ( 2\cos(0) = 2 ).

So, the limit simplifies to:

[ \lim_{x \to 0}To find the limit ( \lim_{x \to 0} \frac{\sin(2x)}{\ln(x+1)} ) using L'Hôpital's Rule, we can differentiate the numerator and denominator separately with respect to ( x ) until we get a determinate form (0/0 or ∞/∞), and then evaluate the limit.

1. Differentiate the numerator and denominator: [ \lim_{x \to 0} \frac{\sin(2x)}{\ln(x+1)} ] [ = \lim_{x \to 0} \frac{d}{dx}\left(\frac{\sin(2x)}{\ln(x+1)}\right) ] [ = \lim_{x \to 0} \frac{2\cos(2x)}{\frac{1}{x+1}} ]

2. Simplify the expression: [ = \lim_{x \to 0} 2\cos(2x)(x+1) ]

3. Evaluate the limit as ( x \to 0 ): [ = 2\cosTo find the limit ( \lim_{x \to 0} \frac{\sin(2x)}{\ln(x+1)} ) using L'Hôpital's Rule, we need to differentiate the numerator and the denominator separately with respect to ( x ) until we reach a determinate form.

4. Differentiate the numerator: ( \frac{d}{dx} \sin(2x) = 2\cos(2x) ).

5. Differentiate the denominator: ( \frac{d}{dx} \ln(x+1) = \frac{1}{x+1} ).

Now, we evaluate the limit of the ratio of the derivatives:

[ \lim_{x \to 0} \frac{2\cos(2x)}{\frac{1}{x+1}} ]

At ( x = 0 ), the denominator becomes ( \frac{1}{1} = 1 ) and the numerator becomes ( 2\cos(0) = 2 ).

So, the limit simplifies to:

[ \lim_{x \to 0} To find the limit ( \lim_{x \to 0} \frac{\sin(2x)}{\ln(x+1)} ) using L'Hôpital's Rule, we can differentiate the numerator and denominator separately with respect to ( x ) until we get a determinate form (0/0 or ∞/∞), and then evaluate the limit.

1. Differentiate the numerator and denominator: [ \lim_{x \to 0} \frac{\sin(2x)}{\ln(x+1)} ] [ = \lim_{x \to 0} \frac{d}{dx}\left(\frac{\sin(2x)}{\ln(x+1)}\right) ] [ = \lim_{x \to 0} \frac{2\cos(2x)}{\frac{1}{x+1}} ]

2. Simplify the expression: [ = \lim_{x \to 0} 2\cos(2x)(x+1) ]

3. Evaluate the limit as ( x \to 0 ): [ = 2\cos(To find the limit ( \lim_{x \to 0} \frac{\sin(2x)}{\ln(x+1)} ) using L'Hôpital's Rule, we need to differentiate the numerator and the denominator separately with respect to ( x ) until we reach a determinate form.

4. Differentiate the numerator: ( \frac{d}{dx} \sin(2x) = 2\cos(2x) ).

5. Differentiate the denominator: ( \frac{d}{dx} \ln(x+1) = \frac{1}{x+1} ).

Now, we evaluate the limit of the ratio of the derivatives:

[ \lim_{x \to 0} \frac{2\cos(2x)}{\frac{1}{x+1}} ]

At ( x = 0 ), the denominator becomes ( \frac{1}{1} = 1 ) and the numerator becomes ( 2\cos(0) = 2 ).

So, the limit simplifies to:

[ \lim_{x \to 0} 2(xTo find the limit ( \lim_{x \to 0} \frac{\sin(2x)}{\ln(x+1)} ) using L'Hôpital's Rule, we can differentiate the numerator and denominator separately with respect to ( x ) until we get a determinate form (0/0 or ∞/∞), and then evaluate the limit.

1. Differentiate the numerator and denominator: [ \lim_{x \to 0} \frac{\sin(2x)}{\ln(x+1)} ] [ = \lim_{x \to 0} \frac{d}{dx}\left(\frac{\sin(2x)}{\ln(x+1)}\right) ] [ = \lim_{x \to 0} \frac{2\cos(2x)}{\frac{1}{x+1}} ]

2. Simplify the expression: [ = \lim_{x \to 0} 2\cos(2x)(x+1) ]

3. Evaluate the limit as ( x \to 0 ): [ = 2\cos(0)(To find the limit ( \lim_{x \to 0} \frac{\sin(2x)}{\ln(x+1)} ) using L'Hôpital's Rule, we need to differentiate the numerator and the denominator separately with respect to ( x ) until we reach a determinate form.

4. Differentiate the numerator: ( \frac{d}{dx} \sin(2x) = 2\cos(2x) ).

5. Differentiate the denominator: ( \frac{d}{dx} \ln(x+1) = \frac{1}{x+1} ).

Now, we evaluate the limit of the ratio of the derivatives:

[ \lim_{x \to 0} \frac{2\cos(2x)}{\frac{1}{x+1}} ]

At ( x = 0 ), the denominator becomes ( \frac{1}{1} = 1 ) and the numerator becomes ( 2\cos(0) = 2 ).

So, the limit simplifies to:

[ \lim_{x \to 0} 2(x+To find the limit ( \lim_{x \to 0} \frac{\sin(2x)}{\ln(x+1)} ) using L'Hôpital's Rule, we can differentiate the numerator and denominator separately with respect to ( x ) until we get a determinate form (0/0 or ∞/∞), and then evaluate the limit.

1. Differentiate the numerator and denominator: [ \lim_{x \to 0} \frac{\sin(2x)}{\ln(x+1)} ] [ = \lim_{x \to 0} \frac{d}{dx}\left(\frac{\sin(2x)}{\ln(x+1)}\right) ] [ = \lim_{x \to 0} \frac{2\cos(2x)}{\frac{1}{x+1}} ]

2. Simplify the expression: [ = \lim_{x \to 0} 2\cos(2x)(x+1) ]

3. Evaluate the limit as ( x \to 0 ): [ = 2\cos(0)(0+To find the limit ( \lim_{x \to 0} \frac{\sin(2x)}{\ln(x+1)} ) using L'Hôpital's Rule, we need to differentiate the numerator and the denominator separately with respect to ( x ) until we reach a determinate form.

4. Differentiate the numerator: ( \frac{d}{dx} \sin(2x) = 2\cos(2x) ).

5. Differentiate the denominator: ( \frac{d}{dx} \ln(x+1) = \frac{1}{x+1} ).

Now, we evaluate the limit of the ratio of the derivatives:

[ \lim_{x \to 0} \frac{2\cos(2x)}{\frac{1}{x+1}} ]

At ( x = 0 ), the denominator becomes ( \frac{1}{1} = 1 ) and the numerator becomes ( 2\cos(0) = 2 ).

So, the limit simplifies to:

[ \lim_{x \to 0} 2(x+1To find the limit ( \lim_{x \to 0} \frac{\sin(2x)}{\ln(x+1)} ) using L'Hôpital's Rule, we can differentiate the numerator and denominator separately with respect to ( x ) until we get a determinate form (0/0 or ∞/∞), and then evaluate the limit.

1. Differentiate the numerator and denominator: [ \lim_{x \to 0} \frac{\sin(2x)}{\ln(x+1)} ] [ = \lim_{x \to 0} \frac{d}{dx}\left(\frac{\sin(2x)}{\ln(x+1)}\right) ] [ = \lim_{x \to 0} \frac{2\cos(2x)}{\frac{1}{x+1}} ]

2. Simplify the expression: [ = \lim_{x \to 0} 2\cos(2x)(x+1) ]

3. Evaluate the limit as ( x \to 0 ): [ = 2\cos(0)(0+1To find the limit ( \lim_{x \to 0} \frac{\sin(2x)}{\ln(x+1)} ) using L'Hôpital's Rule, we need to differentiate the numerator and the denominator separately with respect to ( x ) until we reach a determinate form.

4. Differentiate the numerator: ( \frac{d}{dx} \sin(2x) = 2\cos(2x) ).

5. Differentiate the denominator: ( \frac{d}{dx} \ln(x+1) = \frac{1}{x+1} ).

Now, we evaluate the limit of the ratio of the derivatives:

[ \lim_{x \to 0} \frac{2\cos(2x)}{\frac{1}{x+1}} ]

At ( x = 0 ), the denominator becomes ( \frac{1}{1} = 1 ) and the numerator becomes ( 2\cos(0) = 2 ).

So, the limit simplifies to:

[ \lim_{x \to 0} 2(x+1)To find the limit ( \lim_{x \to 0} \frac{\sin(2x)}{\ln(x+1)} ) using L'Hôpital's Rule, we can differentiate the numerator and denominator separately with respect to ( x ) until we get a determinate form (0/0 or ∞/∞), and then evaluate the limit.

1. Differentiate the numerator and denominator: [ \lim_{x \to 0} \frac{\sin(2x)}{\ln(x+1)} ] [ = \lim_{x \to 0} \frac{d}{dx}\left(\frac{\sin(2x)}{\ln(x+1)}\right) ] [ = \lim_{x \to 0} \frac{2\cos(2x)}{\frac{1}{x+1}} ]

2. Simplify the expression: [ = \lim_{x \to 0} 2\cos(2x)(x+1) ]

3. Evaluate the limit as ( x \to 0 ): [ = 2\cos(0)(0+1)To find the limit ( \lim_{x \to 0} \frac{\sin(2x)}{\ln(x+1)} ) using L'Hôpital's Rule, we need to differentiate the numerator and the denominator separately with respect to ( x ) until we reach a determinate form.

4. Differentiate the numerator: ( \frac{d}{dx} \sin(2x) = 2\cos(2x) ).

5. Differentiate the denominator: ( \frac{d}{dx} \ln(x+1) = \frac{1}{x+1} ).

Now, we evaluate the limit of the ratio of the derivatives:

[ \lim_{x \to 0} \frac{2\cos(2x)}{\frac{1}{x+1}} ]

At ( x = 0 ), the denominator becomes ( \frac{1}{1} = 1 ) and the numerator becomes ( 2\cos(0) = 2 ).

So, the limit simplifies to:

[ \lim_{x \to 0} 2(x+1) = To find the limit ( \lim_{x \to 0} \frac{\sin(2x)}{\ln(x+1)} ) using L'Hôpital's Rule, we can differentiate the numerator and denominator separately with respect to ( x ) until we get a determinate form (0/0 or ∞/∞), and then evaluate the limit.

1. Differentiate the numerator and denominator: [ \lim_{x \to 0} \frac{\sin(2x)}{\ln(x+1)} ] [ = \lim_{x \to 0} \frac{d}{dx}\left(\frac{\sin(2x)}{\ln(x+1)}\right) ] [ = \lim_{x \to 0} \frac{2\cos(2x)}{\frac{1}{x+1}} ]

2. Simplify the expression: [ = \lim_{x \to 0} 2\cos(2x)(x+1) ]

3. Evaluate the limit as ( x \to 0 ): [ = 2\cos(0)(0+1) \To find the limit ( \lim_{x \to 0} \frac{\sin(2x)}{\ln(x+1)} ) using L'Hôpital's Rule, we need to differentiate the numerator and the denominator separately with respect to ( x ) until we reach a determinate form.

4. Differentiate the numerator: ( \frac{d}{dx} \sin(2x) = 2\cos(2x) ).

5. Differentiate the denominator: ( \frac{d}{dx} \ln(x+1) = \frac{1}{x+1} ).

Now, we evaluate the limit of the ratio of the derivatives:

[ \lim_{x \to 0} \frac{2\cos(2x)}{\frac{1}{x+1}} ]

At ( x = 0 ), the denominator becomes ( \frac{1}{1} = 1 ) and the numerator becomes ( 2\cos(0) = 2 ).

So, the limit simplifies to:

[ \lim_{x \to 0} 2(x+1) = 2To find the limit ( \lim_{x \to 0} \frac{\sin(2x)}{\ln(x+1)} ) using L'Hôpital's Rule, we can differentiate the numerator and denominator separately with respect to ( x ) until we get a determinate form (0/0 or ∞/∞), and then evaluate the limit.

1. Differentiate the numerator and denominator: [ \lim_{x \to 0} \frac{\sin(2x)}{\ln(x+1)} ] [ = \lim_{x \to 0} \frac{d}{dx}\left(\frac{\sin(2x)}{\ln(x+1)}\right) ] [ = \lim_{x \to 0} \frac{2\cos(2x)}{\frac{1}{x+1}} ]

2. Simplify the expression: [ = \lim_{x \to 0} 2\cos(2x)(x+1) ]

3. Evaluate the limit as ( x \to 0 ): [ = 2\cos(0)(0+1) ] [ = 2(1)(1) ] [ = 2 ]

So, ( \lim_{x \to 0} \frac{\sin(To find the limit ( \lim_{x \to 0} \frac{\sin(2x)}{\ln(x+1)} ) using L'Hôpital's Rule, we need to differentiate the numerator and the denominator separately with respect to ( x ) until we reach a determinate form.

1. Differentiate the numerator: ( \frac{d}{dx} \sin(2x) = 2\cos(2x) ).
2. Differentiate the denominator: ( \frac{d}{dx} \ln(x+1) = \frac{1}{x+1} ).

Now, we evaluate the limit of the ratio of the derivatives:

[ \lim_{x \to 0} \frac{2\cos(2x)}{\frac{1}{x+1}} ]

At ( x = 0 ), the denominator becomes ( \frac{1}{1} = 1 ) and the numerator becomes ( 2\cos(0) = 2 ).

So, the limit simplifies to:

[ \lim_{x \to 0} 2(x+1) = 2 ]

To find the limit ( \lim_{x \to 0} \frac{\sin(2x)}{\ln(x+1)} ) using L'Hôpital's Rule, we can differentiate the numerator and denominator separately with respect to ( x ) until we get a determinate form (0/0 or ∞/∞), and then evaluate the limit.

1. Differentiate the numerator and denominator: [ \lim_{x \to 0} \frac{\sin(2x)}{\ln(x+1)} ] [ = \lim_{x \to 0} \frac{d}{dx}\left(\frac{\sin(2x)}{\ln(x+1)}\right) ] [ = \lim_{x \to 0} \frac{2\cos(2x)}{\frac{1}{x+1}} ]

2. Simplify the expression: [ = \lim_{x \to 0} 2\cos(2x)(x+1) ]

3. Evaluate the limit as ( x \to 0 ): [ = 2\cos(0)(0+1) ] [ = 2(1)(1) ] [ = 2 ]

So, ( \lim_{x \to 0} \frac{\sin(2To find the limit ( \lim_{x \to 0} \frac{\sin(2x)}{\ln(x+1)} ) using L'Hôpital's Rule, we need to differentiate the numerator and the denominator separately with respect to ( x ) until we reach a determinate form.

1. Differentiate the numerator: ( \frac{d}{dx} \sin(2x) = 2\cos(2x) ).
2. Differentiate the denominator: ( \frac{d}{dx} \ln(x+1) = \frac{1}{x+1} ).

Now, we evaluate the limit of the ratio of the derivatives:

[ \lim_{x \to 0} \frac{2\cos(2x)}{\frac{1}{x+1}} ]

At ( x = 0 ), the denominator becomes ( \frac{1}{1} = 1 ) and the numerator becomes ( 2\cos(0) = 2 ).

So, the limit simplifies to:

[ \lim_{x \to 0} 2(x+1) = 2 ]

Therefore, (To find the limit ( \lim_{x \to 0} \frac{\sin(2x)}{\ln(x+1)} ) using L'Hôpital's Rule, we can differentiate the numerator and denominator separately with respect to ( x ) until we get a determinate form (0/0 or ∞/∞), and then evaluate the limit.

1. Differentiate the numerator and denominator: [ \lim_{x \to 0} \frac{\sin(2x)}{\ln(x+1)} ] [ = \lim_{x \to 0} \frac{d}{dx}\left(\frac{\sin(2x)}{\ln(x+1)}\right) ] [ = \lim_{x \to 0} \frac{2\cos(2x)}{\frac{1}{x+1}} ]

2. Simplify the expression: [ = \lim_{x \to 0} 2\cos(2x)(x+1) ]

3. Evaluate the limit as ( x \to 0 ): [ = 2\cos(0)(0+1) ] [ = 2(1)(1) ] [ = 2 ]

So, ( \lim_{x \to 0} \frac{\sin(2xTo find the limit ( \lim_{x \to 0} \frac{\sin(2x)}{\ln(x+1)} ) using L'Hôpital's Rule, we need to differentiate the numerator and the denominator separately with respect to ( x ) until we reach a determinate form.

1. Differentiate the numerator: ( \frac{d}{dx} \sin(2x) = 2\cos(2x) ).
2. Differentiate the denominator: ( \frac{d}{dx} \ln(x+1) = \frac{1}{x+1} ).

Now, we evaluate the limit of the ratio of the derivatives:

[ \lim_{x \to 0} \frac{2\cos(2x)}{\frac{1}{x+1}} ]

At ( x = 0 ), the denominator becomes ( \frac{1}{1} = 1 ) and the numerator becomes ( 2\cos(0) = 2 ).

So, the limit simplifies to:

[ \lim_{x \to 0} 2(x+1) = 2 ]

Therefore, ( \lim_{x \to 0} \frac{\sin(To find the limit ( \lim_{x \to 0} \frac{\sin(2x)}{\ln(x+1)} ) using L'Hôpital's Rule, we can differentiate the numerator and denominator separately with respect to ( x ) until we get a determinate form (0/0 or ∞/∞), and then evaluate the limit.

1. Differentiate the numerator and denominator: [ \lim_{x \to 0} \frac{\sin(2x)}{\ln(x+1)} ] [ = \lim_{x \to 0} \frac{d}{dx}\left(\frac{\sin(2x)}{\ln(x+1)}\right) ] [ = \lim_{x \to 0} \frac{2\cos(2x)}{\frac{1}{x+1}} ]

2. Simplify the expression: [ = \lim_{x \to 0} 2\cos(2x)(x+1) ]

3. Evaluate the limit as ( x \to 0 ): [ = 2\cos(0)(0+1) ] [ = 2(1)(1) ] [ = 2 ]

So, ( \lim_{x \to 0} \frac{\sin(2x)}To find the limit ( \lim_{x \to 0} \frac{\sin(2x)}{\ln(x+1)} ) using L'Hôpital's Rule, we need to differentiate the numerator and the denominator separately with respect to ( x ) until we reach a determinate form.

1. Differentiate the numerator: ( \frac{d}{dx} \sin(2x) = 2\cos(2x) ).
2. Differentiate the denominator: ( \frac{d}{dx} \ln(x+1) = \frac{1}{x+1} ).

Now, we evaluate the limit of the ratio of the derivatives:

[ \lim_{x \to 0} \frac{2\cos(2x)}{\frac{1}{x+1}} ]

At ( x = 0 ), the denominator becomes ( \frac{1}{1} = 1 ) and the numerator becomes ( 2\cos(0) = 2 ).

So, the limit simplifies to:

[ \lim_{x \to 0} 2(x+1) = 2 ]

Therefore, ( \lim_{x \to 0} \frac{\sin(2x)}{\ln(x+1)}To find the limit ( \lim_{x \to 0} \frac{\sin(2x)}{\ln(x+1)} ) using L'Hôpital's Rule, we can differentiate the numerator and denominator separately with respect to ( x ) until we get a determinate form (0/0 or ∞/∞), and then evaluate the limit.

1. Differentiate the numerator and denominator: [ \lim_{x \to 0} \frac{\sin(2x)}{\ln(x+1)} ] [ = \lim_{x \to 0} \frac{d}{dx}\left(\frac{\sin(2x)}{\ln(x+1)}\right) ] [ = \lim_{x \to 0} \frac{2\cos(2x)}{\frac{1}{x+1}} ]

2. Simplify the expression: [ = \lim_{x \to 0} 2\cos(2x)(x+1) ]

3. Evaluate the limit as ( x \to 0 ): [ = 2\cos(0)(0+1) ] [ = 2(1)(1) ] [ = 2 ]

So, ( \lim_{x \to 0} \frac{\sin(2x)}{\To find the limit ( \lim_{x \to 0} \frac{\sin(2x)}{\ln(x+1)} ) using L'Hôpital's Rule, we need to differentiate the numerator and the denominator separately with respect to ( x ) until we reach a determinate form.

1. Differentiate the numerator: ( \frac{d}{dx} \sin(2x) = 2\cos(2x) ).
2. Differentiate the denominator: ( \frac{d}{dx} \ln(x+1) = \frac{1}{x+1} ).

Now, we evaluate the limit of the ratio of the derivatives:

[ \lim_{x \to 0} \frac{2\cos(2x)}{\frac{1}{x+1}} ]

At ( x = 0 ), the denominator becomes ( \frac{1}{1} = 1 ) and the numerator becomes ( 2\cos(0) = 2 ).

So, the limit simplifies to:

[ \lim_{x \to 0} 2(x+1) = 2 ]

Therefore, ( \lim_{x \to 0} \frac{\sin(2x)}{\ln(x+1)} = 2 ) using L'HôpTo find the limit ( \lim_{x \to 0} \frac{\sin(2x)}{\ln(x+1)} ) using L'Hôpital's Rule, we can differentiate the numerator and denominator separately with respect to ( x ) until we get a determinate form (0/0 or ∞/∞), and then evaluate the limit.

1. Differentiate the numerator and denominator: [ \lim_{x \to 0} \frac{\sin(2x)}{\ln(x+1)} ] [ = \lim_{x \to 0} \frac{d}{dx}\left(\frac{\sin(2x)}{\ln(x+1)}\right) ] [ = \lim_{x \to 0} \frac{2\cos(2x)}{\frac{1}{x+1}} ]

2. Simplify the expression: [ = \lim_{x \to 0} 2\cos(2x)(x+1) ]

3. Evaluate the limit as ( x \to 0 ): [ = 2\cos(0)(0+1) ] [ = 2(1)(1) ] [ = 2 ]

So, ( \lim_{x \to 0} \frac{\sin(2x)}{\lnTo find the limit ( \lim_{x \to 0} \frac{\sin(2x)}{\ln(x+1)} ) using L'Hôpital's Rule, we need to differentiate the numerator and the denominator separately with respect to ( x ) until we reach a determinate form.

1. Differentiate the numerator: ( \frac{d}{dx} \sin(2x) = 2\cos(2x) ).
2. Differentiate the denominator: ( \frac{d}{dx} \ln(x+1) = \frac{1}{x+1} ).

Now, we evaluate the limit of the ratio of the derivatives:

[ \lim_{x \to 0} \frac{2\cos(2x)}{\frac{1}{x+1}} ]

At ( x = 0 ), the denominator becomes ( \frac{1}{1} = 1 ) and the numerator becomes ( 2\cos(0) = 2 ).

So, the limit simplifies to:

[ \lim_{x \to 0} 2(x+1) = 2 ]

Therefore, ( \lim_{x \to 0} \frac{\sin(2x)}{\ln(x+1)} = 2 ) using L'Hôpital's Rule.To find the limit ( \lim_{x \to 0} \frac{\sin(2x)}{\ln(x+1)} ) using L'Hôpital's Rule, we can differentiate the numerator and denominator separately with respect to ( x ) until we get a determinate form (0/0 or ∞/∞), and then evaluate the limit.

1. Differentiate the numerator and denominator: [ \lim_{x \to 0} \frac{\sin(2x)}{\ln(x+1)} ] [ = \lim_{x \to 0} \frac{d}{dx}\left(\frac{\sin(2x)}{\ln(x+1)}\right) ] [ = \lim_{x \to 0} \frac{2\cos(2x)}{\frac{1}{x+1}} ]

2. Simplify the expression: [ = \lim_{x \to 0} 2\cos(2x)(x+1) ]

3. Evaluate the limit as ( x \to 0 ): [ = 2\cos(0)(0+1) ] [ = 2(1)(1) ] [ = 2 ]

So, ( \lim_{x \to 0} \frac{\sin(2x)}{\ln(x+1)} = 2 ) using L'Hôpital's Rule.