# How can I find the limit of a function that is inside a limit?

For example:

Given $\lim_{x\to 4} \frac{5xf(x)−1}{x−4}= 8$, find $\lim_{x\to 4} f(x)$.

A good hint would be appreciated!

• Hint: the denominator tends to zero. What must the numerator tend to if the limit of the quotient exists? – symplectomorphic Aug 27 '16 at 15:42
• Formatting tips here. – Em. Aug 27 '16 at 15:43
• @symplectomorphic Thank you :) – Smebbs Aug 27 '16 at 15:48

Note that $$\lim_{x\rightarrow 4}(5xf(x)−1) =\lim_{x\rightarrow 4}(x-4)\lim_{x\rightarrow 4}\frac{5xf(x)−1}{x−4}=0.$$ So $$0=\lim_{x\rightarrow 4}(5xf(x)−1)=\left(\lim_{x\rightarrow 4}5x\right)\left(\lim_{x\rightarrow 4}f(x)\right)-1$$