How to find the limit of sin(x)/x as x approaches zero?

For example, the problem is sin (2x)/ 8x, how to find the limit as x approaches to zero?

Question

How to find the limit of sin(x)/x as x approaches zero?

For example, the problem is sin (2x)/ 8x, how to find the limit as x approaches to zero?

anonymous · Answer

will it be 1/4 since the limit of sin(x)/x is equal to 1?

displayerror · Answer

We know that $$\lim_{x \rightarrow 0} \frac{\sin x}{x} = 1$$

Looking at the given problem
$$\lim_{x \rightarrow 0} \frac{\sin 2x}{8x} $$

In order to get the problem to look like the form sin x / x, we need the denominator of the fraction to equal 2x. Setting the bottom to 2x, the limit looks like this:
$$n \  \lim_{x \rightarrow 0} \frac{\sin 2x}{2x} $$
In order for the above to be true, we would have to multiply the limit by some number $n$. That's what you have to find.

displayerror · Answer

Yeah, it works out to be 1/4.

anonymous · Answer

ah okay thanks! 

one more question, why is it that $\lim_{x \rightarrow 0} \frac{\sin x}{x} = 1$ ? 
Is there a proof that it is equal to 1?

anonymous · Answer

I just know about it but I never learned the reason behind it