What is the Pinching/Sandwhich/Squeeze Theorem in Calculus?

Question

Can someone please provide a detailed description of what the pinching theorom is. Perhaps using

lim xsin(1/x)=0
x-->0

Answer 1

Squeeze theorem:

If you know that something is always between to points (inclusively) and that those two points eventually come together (or their limit comes together), then you know that the point is equal to the limits of the others.

For your example:

sin(u)≤1 for all u. Thus sin(1/x)≤1 for all x≠0
Likewise sin(1/x)≥-1 for all x≠0.

Thus -x≤xsin(1/x)≤x for all x>0. Since -x-->0 as x--->0+ and x--->0 as x-->0+, by the squeeze theorem xsin(1/x)-->0 as x-->0+

This can be repeated to show that xsin(1/x)-->0 as x-->0- also, and thus the right and left limits equal and the limit is zero.

Answer 2

Pinch your girlfriends butt, eat a sandwich, then squeeze out a duece!! LOL I'm sorry, I had to. It's one of those days.