Math 150: Calculus III
Professor Steven Miller
Williams College
sjm1@williams.edu

Example of the Squeeze Theorem

Manipulate[Plot[Abs[x]^exponent Sin[1/x], {x,-r,r}], {exponent,0,2},{r,4,.0004}]

The squeeze theorem is a great way to compute limits.
If we know a(n) <= b(n) <= c(n) and the limit as n goes to infinity of a(n) and c(n) is L, then L is also the limit of b(n) as n goes to infinity. We can also do this for functions: if a(x) <= b(x) <= c(x) and if a(x) and c(x) have limit L as x goes to some point x_0, then that is the limit of b(x) as x goes to x_0.

The example above plots |x|^r sin(1/x) for various exponents.
For example, when r = 2 our function oscillates between the parabolas y = x^2 and y = -x^2, and as x tends to zero we see the limit is zero.