Let $f(x) = 0$ if $x$ is rational and $f(x)=1$ if $x$ is irrational. Prove that $\lim_{x\to a} f(x)$ does not exist for any $a$.
Proof attempt:
Let $a\in\mathbb{R}$. The only two possible values for a limit of $\lim_{x\to a}f(x)$ are $0,1$, since otherwise there will always be a constant gap between the valules of $f$ and the limit, and hence $f$ would not get arbitrarily small.
However, $\lim_{x\to a}f(x) \not = 1$ for the following reason: Set $\epsilon = 1/2$ and let $\delta > 0$. Then there exists a rational number $x_0$ in $0<|x_0-a|<\delta$ by the density of the rationals. However, $|f(x_0)-1| = |0-1| = 1 \not < 1/2$, and so $\lim_{x\to a}f(x) \not = 1$.
Also, $\lim_{x\to a}f(x) \not = 0$ for the following reason: Set $\epsilon = 1/2$ and let $\delta > 0$. Then there exists a irrational number $x_0$ in $0<|x_0-a|<\delta$ by the density of the irrationals. However, $|f(x_0)-0| = |1-0| = 1 \not < 1/2$, and so $\lim_{x\to a}f(x) \not = 0$.