Existence of a limit for a rational/irrational-valued function

Choose values of $x$ approaching 0 that are all rational, such as $x = 0.1, 0.01, 0.001, \ldots$. For every one of these, $t(x) = 2$, so along this approach the values seem to head toward $2$.

Now choose values of $x$ approaching 0 that are all irrational, such as $x = 0.1\sqrt{2}, 0.01\sqrt{2}, 0.001\sqrt{2}, \ldots$. For every one of these, $t(x) = -2$, so along this approach the values head toward $-2$ instead.

Since we can get arbitrarily close to $x=0$ while $t(x)$ stays exactly at $2$, and also get arbitrarily close to $x=0$ while $t(x)$ stays exactly at $-2$, the function does not approach a single value as $x \to 0$. It oscillates between $2$ and $-2$ no matter how close $x$ gets to $0$, so the limit fails to exist.