# How do you find the limit of #x/sinx# as x approaches 0?

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Let

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We need to know the important trigonometric limit:

#lim_(xrarr0)sinx/x=1#

So, we see that:

#lim_(xrarr0)x/sinx=lim_(xrarr0)(sinx/x)^-1=(lim_(xrarr0)sinx/x)^-1=1^-1=1#

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Use the squeeze theorem;

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We can use the squeeze theorem (or sandwich theorem), which states that if

So the challenge is to find upper and lower bounds for

Consider:

We have:

{Area of

Multiplying by

Hence, by the squeeze theorem,

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If we try to calculate the limit directly, we can see that is an indeterminate form:

To solve it, we can apply the L'Hôpital's rule:

Given two functions

If

So we have:

**NOTE**

The question was posted in **"Determining Limits Algebraically"** , so the use of L'Hôpital's rule is **NOT** a suitable method to solve the problem. Therefore this solution is invalid.

*ANSWER TO THE NOTE*

This limit can not be solved using only algebraic concepts as the function

The answer above that uses the limit

I think the person who wrote the first note confuses the term "algebraic" in the expression "Determining Limits algebraically" in the true meaning of the word algebraic, regardless of that expression usually means "Determination of limits analytically", ie, by calculation and not graphically, numerically (using approximations) or intuitive.

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