How to find functions in order to apply Squeeze Rule for continuous functions

Question

In our course we were introduced to the the Squeeze Rule for continuous functions.

An example was given where the Squeeze Rule was used to prove that the following function is continuous at point $0$.

$$ f(x) = \begin{cases} x^2 \sin(1/x) & x \not = 0 \\ 0 & x = 0 \end{cases} $$

Looking at a graph of $f(x)$, $g(x) = x^2$ and $h(x) = -x^2$ makes it pretty obvious that we could use $g, h$ together with the Squeeze Rule.

But in a test they won't be so nice as to provide us with visual aids, also the use of a graphing calculator is forbidden.

Are there any common techniques/clues we can apply if we are only given $f$ and it's graph (this they will provide in an examination) to find $g, h$ to apply the Squeeze Rule, proving continuity of $f$? Based on the definition of $f$ above alone I doubt I would have thought of $g(x) = x^2$ and $h(x) = -x^2$ straight away.

Answer 1

One of the most common problems includes a sine or cosine function. Those functions come with inequalities built in: $-1 \leq \sin x \leq 1$ and likewise for cosine.

Another common example is to have a rational function, and compare it to a simpler one that is larger and a simpler one that is smaller. So for example, given $\frac{x+3}{x^2 + 4}$, you might pick: $$\frac{x}{x^2 + 4} \leq \frac{x+3}{x^2 + 4} \leq \frac{2x}{x^2 + 4}$$ (for $x>3$) assuming you are not approaching $0$; other choices might be suitable for different limit points.

It is very similar to comparing power series with the comparison tests, and we make very similar choices to work those problems.

Answer 2

The relevant part is the expression $x^2\sin(1/x)$. The factor $\sin(1/x)$ is always somewhere between $-1$ and $1$. So that expression will always be somewhere between $x^2$ and $-x^2$. And trying it out, you'll then see that those two functions do work for the squeeze theorem.

Generally, if you have a product $u(x)v(x)$ where $v(x)$ has the lower bound $a$ and upper bound $b$, then it's a good idea to try $au(x)$ and $bu(x)$ as bounds for the squeeze theorem.