How Can You Use the Squeeze Theorem with Non-Trig Functions?

[Dr. Seafood]

The usefulness of the squeeze theorem is almost exclusively (in my experience) presented by a trig function, since the elementary functions sine and cosine are bounded; namely -1 ≤ sin x, cos x ≤ 1 for all x.

I'm looking for an example of the squeeze theorem involving elementary real-valued functions aside from trig functions. Namely, functions P, Q and F such that P(x) ≤ F(x) ≤ Q(x) for all x near some value x₀ and P and Q have the same limits at x₀. Of course it is not necessary that F(x₀) be defined, in fact it might be a better example if it is not immediately clear how to evaluate the limit of F at x₀.

Probably, P and Q would be polynomials, and F is a rational function discontinuous at x₀.

(If you're wondering, I'm a math tutor and I want to present an example of the squeeze theorem but I haven't introduced trig functions yet. In fact I will use the squeeze theorem to develop the derivatives of the sine and cosine functions.)

 

[wisvuze]

Well something like the squeeze theorem is implicitly used when thinking about the "comparison test" for non-negative sequences. An intuitive depiction of the squeeze theorem could involve two non negative sequences, one of which remains an upper bound for the other

 

[Dr. Seafood]

^ Very true, thanks. I'm looking to demonstrate the squeeze theorem in the context of real-valued functions, not sequences.

Hmm, but that gives me the idea to show a result similar to the sqz thm: if f(x) ≤ g(x) for (large) x and the limit of g(x) is L, then the limit of f(x) is also L. I can give an easy example using this, thanks.