Answer
$f'(-1)$, where $f(x)= x^{3}$
Work Step by Step
The derivative of $f(x)$ at a point $a$ is defined as
$f'(a)=\lim\limits_{x \to a}\frac{f(x)-f(a)}{x-a}$
Comparing $\lim\limits_{x\to -1}\frac{x^{3}+1}{x+1}$ with the above definition, we get
$a=-1$
$f(x)-f(a)=x^{3}+1$
$f(x)=x^{3}$ and
$f(a)= f(-1)= (-1)^{3}=-1$
Therefore, the given limit can be expressed as the derivative of $f(x)= x^{3}$ at $x=-1$
