One-sided infinite limits at a vertical asymptote

As $x \to 2^-$, the values of $x$ are slightly less than 2, so $x - 2$ is a small negative number approaching 0 (for example, $x = 1.99$ gives $x-2 = -0.01$). Dividing 1 by a small negative number gives a large negative number, so $f(x) \to -\infty$.

As $x \to 2^+$, the values of $x$ are slightly greater than 2, so $x-2$ is a small positive number approaching 0 (for example, $x = 2.01$ gives $x-2 = 0.01$). Dividing 1 by a small positive number gives a large positive number, so $f(x) \to +\infty$. Since the one-sided limits are infinite, this confirms $x=2$ is a vertical asymptote.