Q.Evaluate $\lim_{x\to1}\big(2x^3-x^2+4x-1\big)$.

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

numerator and denominator by the Factor Theorem; cancelling this common factor and only then

substituting $x=a$ resolves the limit. A key standard result following from this factoring

technique is $\lim_{x\to a}(x^n-a^n)/(x-a)=na^{n-1}$, proved by factoring the numerator into $n$

terms each of which tends to $a^{n-1}$.