# How do you find the limit of #(sin(x)/3x) # as x approaches 0 using l'hospital's rule?

See explanation below.

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To find the limit of (sin(x)/3x) as x approaches 0 using L'Hospital's rule, we can differentiate the numerator and denominator separately.

Differentiating the numerator, we get cos(x). Differentiating the denominator, we get 3.

Now, we can evaluate the limit of the derivative of the numerator divided by the derivative of the denominator as x approaches 0.

The limit of cos(x)/3 as x approaches 0 is equal to cos(0)/3, which simplifies to 1/3.

Therefore, the limit of (sin(x)/3x) as x approaches 0 using L'Hospital's rule is 1/3.

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To find the limit of ( \frac{\sin(x)}{3x} ) as ( x ) approaches 0 using L'Hôpital's Rule:

- Take the derivative of the numerator and the derivative of the denominator separately.
- Evaluate both derivatives at ( x = 0 ).
- Take the limit of the ratio of the derivatives as ( x ) approaches 0.

Derivative of ( \sin(x) ) with respect to ( x ) is ( \cos(x) ). Derivative of ( 3x ) with respect to ( x ) is ( 3 ).

At ( x = 0 ), ( \sin(0) = 0 ) and ( 3(0) = 0 ).

Thus, the limit becomes ( \lim_{x \to 0} \frac{\cos(x)}{3} ).

At ( x = 0 ), ( \cos(0) = 1 ).

So, the limit is ( \frac{1}{3} ).

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