# How to find the limit 5(1-cos2x)/sin2x as x approach 0 from the left ?

To assess:

Now think about the well-known boundaries:

and we possess:

graph{sin(2x) [-10, 10, -5, 5]}/(5(1-cos(2x))

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Because the Limit is real, we have

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To find the limit of ( \frac{5(1 - \cos(2x))}{\sin(2x)} ) as ( x ) approaches ( 0 ) from the left:

- Replace ( x ) with ( 0 ) in the expression.
- Simplify the expression.
- Evaluate the limit.

Step 1: Substitute ( x = 0 ) into the expression: [ \frac{5(1 - \cos(2 \cdot 0))}{\sin(2 \cdot 0)} ]

Step 2: Simplify the expression: [ \frac{5(1 - \cos(0))}{\sin(0)} ] [ = \frac{5(1 - 1)}{0} ] [ = \frac{5 \cdot 0}{0} ]

Step 3: Evaluate the limit: [ \lim_{x \to 0^-} \frac{5(1 - \cos(2x))}{\sin(2x)} = \lim_{x \to 0^-} \frac{0}{0} ]

When you have ( \frac{0}{0} ), it indicates that the expression is indeterminate. To evaluate this type of limit, you can use L'Hôpital's Rule, which states that if the limit of the quotient of two functions approaches ( \frac{0}{0} ) or ( \frac{\infty}{\infty} ), then you can differentiate the numerator and the denominator separately and take the limit again.

[ \lim_{x \to 0^-} \frac{5(1 - \cos(2x))}{\sin(2x)} = \lim_{x \to 0^-} \frac{5(2\sin(2x))}{2\cos(2x)} ]

[ = \lim_{x \to 0^-} \frac{5\sin(2x)}{\cos(2x)} ]

Now, substitute ( x = 0 ) into this expression:

[ = \frac{5\sin(0)}{\cos(0)} ] [ = \frac{5 \cdot 0}{1} ] [ = 0 ]

So, the limit of ( \frac{5(1 - \cos(2x))}{\sin(2x)} ) as ( x ) approaches ( 0 ) from the left is ( 0 ).

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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