How would I explain what scenarios the squeeze theorem should be used in?

[MurdocJensen]

This is what I want to say:

The squeeze theorem may be used when direct substitution and factoring (or simplification of any sort) doesn't help in finding a limit.

An example would be lim x->0 of x²sin(pi/x). Limit laws wouldn't work and we can't simplify the expression. What we can do is recognize -1<= sin(pi/x)<= 1, so -(x²) <= x²sin(pi/x) <= x². According to the squeeze theorem, if the function in question is bounded by two other function at x near a, and the lmits of these bounding functions equal each other at some a, then the limit of the bounded function also takes that value at a.

 

[fresh_42]

https://en.wikipedia.org/wiki/Squeeze_theorem has even proofs.