I'm trying to do an exercise but I honestly don't know what to do to solve it. I know this will be a very basic question, but I'm learning limits.
The exercise is:
Given:
$$\lim\limits_{x \to 2} \frac{f(x) - 6}{x^2-4} = 5$$
Find: $$\lim\limits_{x \to 2} f(x)$$
I tried separating the limit but I didn't get anything from it.
If you could please help me understand what to do, I'd be greatful (I need to understand, not just the answer).
Since$$\lim_{x\to2}\frac{f(x)-6}{x^2-4}=5,$$you have\begin{align}\lim_{x\to2}f(x)-6&=\lim_{x\to2}\left(\frac{f(x)-6}{x^2-4}(x^2-4)\right)\\&=\lim_{x\to2}\frac{f(x)-6}{x^2-4}\times\lim_{x\to2}(x^2-4)\\&=5\times0\\&=0.\end{align}So, $\lim_{x\to2}f(x)=6$.