Answer
$\lim_{x\to2}\dfrac{x^{4}-16}{x-2}=32$
Work Step by Step
$\lim_{x\to2}\dfrac{x^{4}-16}{x-2}$
Try to evaluate the limit using direct substitution:
$\lim_{x\to2}\dfrac{x^{4}-16}{x-2}=\dfrac{(2)^{4}-16}{2-2}=\dfrac{16-16}{2-2}=\dfrac{0}{0}$ Indeterminate form
The limit could not be evaluated by direct substitution. Factor the function completely and simplify:
$\lim_{x\to2}\dfrac{x^{4}-16}{x-2}=\lim_{x\to2}\dfrac{(x^{2}-4)(x^{2}+4)}{x-2}=...$
$...=\lim_{x\to2}\dfrac{(x-2)(x+2)(x^{2}+4)}{x-2}=$
$...=\lim_{x\to2}(x+2)(x^{2}+4)=...$
Try to evaluate the limit using direct substitution again:
$...=\lim_{x\to2}(x+2)(x^{2}+4)=(2+2)[2^{2}+4]=(4)(4+4)=...$
$...=(4)(8)=32$
