Chapter 2 - Section 2.3 - Calculating Limits Using the Limit Laws - 2.3 Exercises - Page 104: 61

$\lim\limits_{x \to 0}f(x) = 0$

$f(x) = x^2$, if $x$ is rational $f(x) = 0$, if $x$ is irrational Let $g(x) = 0$ and let $h(x) = \vert x \vert $ When $x$ is close to $0$, $g(x) \leq f(x) \leq h(x)$ Also, $\lim\limits_{x \to 0}g(x) = 0$ and $\lim\limits_{x \to 0}h(x) = 0$ Therefore, by the Squeeze Theorem, $\lim\limits_{x \to 0}f(x) = 0$