Simplest explanation of a limit for a calc I student

Question

I had a calculus I student come to my office hours with the following problem: Suppose $\lim_{x\to a}\frac{f(x)}{(x-a)^2}=L$. Find $\lim_{x\to a}\frac{f(x)}{x-a}$. I sifted through my real analysis bag of tricks to find a solution for them but unfortunately it seemed to go beyond the scope of their class. If anyone has any pedagogical advice on how to explain this problem to a calculus I student, that would be great!

Answer 1

This is just an elementary limit law. You've assumed that $\lim_{x \to a} \frac{f(x)}{(x-a)^2}$ exists and is equal to $L$. Then $$ \lim_{x \to a} \frac{f(x)}{x-a} = \lim_{x \to a} \left(\frac{f(x)}{(x-a)^2} \cdot (x-a) \right)= \lim_{x \to a} \frac{f(x)}{(x-a)^2} \cdot \lim_{x \to a}(x-a) = L\cdot 0 =0. $$