Limits of Polynomial and Rational Functions

Limits of polynomials. A polynomial $p(x) = c_0+c_1x+c_2x^2+\cdots+c_nx^n$

is built entirely out of the constant function and the identity function $x$, combined only by

addition, subtraction, scalar multiplication and (repeated) multiplication. Since

$\lim_{x\to a}c=c$ and $\lim_{x\to a}x=a$ (Section 2), applying the product rule repeatedly

gives $\lim_{x\to a}x^k=a^k$ for every positive integer $k$, and applying the sum and

scalar-multiple rules to the whole polynomial gives the single most useful fact in this section:

for every polynomial $p$ and every real $a$ -- the limit of a polynomial is found simply by

direct substitution, $x=a$.

Limits of rational functions. A rational function is a quotient of two polynomials,

$f(x)=\dfrac{p(x)}{q(x)}$. By the quotient rule of Section 2, if $q(a)\neq0$,

-- again direct substitution. Two other cases need separate care:

• If $q(a)=0$ but $p(a)\neq0$, the quotient has no finite limit at $a$ (it grows without bound as $x\to a$); this case falls outside the scope of this chapter.
• If both $p(a)=0$ and $q(a)=0$ -- the $0/0$ indeterminate form flagged in Section 2 -- then $(x-a)$ is a common factor of both $p(x)$ and $q(x)$ (a consequence of the Factor Theorem from algebra). Cancel the common factor first, then substitute $x=a$ into what remains; the cancelled function agrees with the original everywhere except at $x=a$ itself, and Section 1 already established that the value AT $a$ never affects the limit.

A standard limit. The following result, used constantly through the rest of this chapter and

into the derivative sections, deserves its own statement and proof.

Theorem. For any positive integer $n$ and any real $a$,

Proof. Factor the numerator using the standard algebraic identity

a sum of exactly $n$ terms. Since $x\neq a$ throughout the limiting process, the factor $(x-a)$

cancels with the denominator, leaving

$\frac{x^n-a^n}{x-a} = x^{n-1}+x^{n-2}a+x^{n-3}a^2+\cdots+xa^{n-2}+a^{n-1}.$ …