We can represent the limit above by a combination of two limits for the rational functions involved.

$\lim_{x\to2} (\frac{1}{x - 2} - \frac{4}{x^2 - 4}) = \lim_{x\to2} \frac{1}{x - 2} - \lim_{x\to2} \frac{4}{x^2 - 4}$

But from close inspection, we can infer that both diverge when x approaches 2, so we can combine them into a single rational function and find the limit.

$\lim_{x\to2} (\frac{1}{x - 2} - \frac{4}{x^2 - 4})$

$\lim_{x\to2} \frac{x - 2}{x^2 - 4}$

$\lim_{x\to2} \frac{x - 2}{(x - 2)(x + 2)}$

$\lim_{x\to2} \frac{1}{x + 2} = \frac{1}{2 + 2} = \frac{1}{4}$