Answer
$\dfrac{3}{8}$
Work Step by Step
We want to find $\lim\limits_{v \to 2}\dfrac{v^3-8}{v^4-16}$, but we can't use the quotient rule for limits because the limit of the denominator as $v$ approaches 2 is zero, and if we try to substitute $v=2$ directly, we get zero in both the numerator and the denominator.
But note that $\dfrac{v^3-8}{v^4-16}=\dfrac{(v-2)(v^2+2v+4)}{(v^2-4)(v^2+4)}=\dfrac{(v-2)(v^2+2v+4)}{(v-2)(v+2)(v^2+4)}=\dfrac{v^2+2v+4}{(v+2)(v^2+4)}$
for all $v \neq 2.$
Thus $\lim\limits_{v \to 2}\dfrac{v^3-8}{v^4-16}=\lim\limits_{v \to 2}\dfrac{v^2+2v+4}{(v+2)(v^2+4)}=\dfrac{4+4+4}{(4)(8)}=\dfrac{12}{4(8)}=\dfrac{3}{8}.$
