How to prove that $\lim\limits_{x\to0}\frac{\tan x}x=1$? How to prove that 
$$\lim\limits_{x\to0}\frac{\tan x}x=1?$$
I'm looking for a method besides L'Hospital's rule.

One way to look at it is to consider an angle subtended by two finite lines, both of magnitude r, where the angle between them is x (we take x to be small). If you draw this out, you can see there are "3 areas" you can consider. One is the area enclosed with a straight line joining the two end points, an arc and lastly considering a right-angled triangle. Sorry I cant provide a diagram, I'm new to maths.stackexchange :)
you get the following result
1/2*r^2sinx < 1/2*r^2x < 1/2*r^2tanx for small x, with simplication we get
sinx < x < tanx divide by tanx yeilds
cosx < x/tanx < 1 taking the limit as x goes to 0, (which we can do as we took x to be small)
we get 1 < x/tanx < 1, by squeeze theorem this tells us the limit of as x >>0 for x/tanx is 1. Now the limit of tans/x as x approaches 0 will be the reciprocal of this. I should mention I am assuming early foundational results regarding limits in an Analysis course. Hence, the limit is 1.
