Examples of squeeze theorem

1. Aug 1, 2011

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.)

2. Aug 1, 2011

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

3. Aug 2, 2011

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.

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