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!

  • 10
    $\begingroup$ Hint: the denominator tends to zero. What must the numerator tend to if the limit of the quotient exists? $\endgroup$ – symplectomorphic Aug 27 '16 at 15:42
  • $\begingroup$ Formatting tips here. $\endgroup$ – Em. Aug 27 '16 at 15:43
  • $\begingroup$ @symplectomorphic Thank you :) $\endgroup$ – 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 $$0=\lim_{x\rightarrow 4}(5xf(x)−1)=\left(\lim_{x\rightarrow 4}5x\right)\left(\lim_{x\rightarrow 4}f(x)\right)-1$$


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