Limit of Sin x/x=1? Proven?

[agapito]

From elementary calculus it is known that

(lim x-->0) ((sin x)/x) = 1.

Is this result equivalent to (lim x-->0) sin x = x ?

If so, how is it proved? Many thanks for all guidance.

 

[HOI]

No, that makes no sense.
The left is a number, the limit of sin(x) as x goes to 0 (which happens to be 0) while the right is a function, x.

What is true is that $\lim_{x\to 0} sin(x)= \lim_{x\to 0} x$ which is simply 0= 0.

 

[agapito]

OK thanks for responding

 

[HOI]

I think I should point out that while $\lim_{x\to a} f(x)= \lim_{x\to a}g(x)$ is a necessary condition for $\lim_{x\to a}\frac{f(x)}{g(x)}= 1$ it is not sufficient.

For example $\lim_{x\to 0} sin(x)= \lim_{x\to 0} x^2= 0$ but $\lim_{x\to 0}\frac{sin(x)}{x^2}$ does not exist.