Q.Evaluate $\lim_{x\to1}\dfrac{x^3-1}{x-1}$ using the standard result $\lim_{x\to a}\dfrac{x^n-a^n}{x-a}=na^{n-1}$.

Because a polynomial $p(x)$ is

built only from constants and powers of $x$ combined by addition and scalar multiplication, the

algebra of limits shows that $\lim_{x\to a}p(x)=p(a)$ for every polynomial and every real $a$ --

its limit is found by direct substitution. The same holds for a rational function $p(x)/q(x)$

whenever the denominator's value $q(a)$ is nonzero. When both $p(a)=0$ and $q(a)=0$, however,

the quotient is an indeterminate $0/0$ form, and $(x-a)$ is a common factor of both the …