05:08
Calculus for Scientists and Engineers: Early Transcendental
Proving that $\lim _{x \rightarrow a} f(x) \neq L$ Use the following definition for the nonexistence of a limit. Assume $f$ is defined for all values of $x$ near a, except possibly at a. We say that $\lim _{x \rightarrow a} f(x) \neq L$ iffor some $\varepsilon>0$ there is no value of $\delta>0$ satisfying the condition $$ |f(x)-L|<\varepsilon \text { whenever } 0<|x-a|<\delta $$
Let
$$
f(x)=\left\{\begin{array}{ll}
0 & \text { if } x \text { is rational } \\
1 & \text { if } x \text { is irrational. }
\end{array}\right.
$$
Prove that $\lim _{x \rightarrow a} f(x)$ does not exist for any value of $a$. (Hint:
Assume $\left.\lim _{x \rightarrow a} f(x)=L \text { for some values of } a \text { and } L \text { and let } \varepsilon=\frac{1}{2} .\right)$