Infinite Limits, Limits at Infinity, and Asymptotes

Infinite Limits and Asymptotes

By the end of this lesson, you will be able to evaluate limits at infinity, identify horizontal and vertical asymptotes, and use end behavior to analyze rational functions.

Vertical Asymptotes

A function has a vertical asymptote at $x = a$ if $\lim_{x\to a^+}f(x) = \pm\infty$ or $\lim_{x\to a^-}f(x) = \pm\infty$.

Cause: the denominator → 0 while the numerator → nonzero.

Example: $f(x) = \frac{1}{(x-2)^2}$: as $x\to 2$, denominator $\to 0^+$, so $f(x) \to +\infty$. Vertical asymptote at $x = 2$.

Careful: if numerator also → 0, it's not necessarily a vertical asymptote — it might be a removable discontinuity (hole). Compare:

• $\frac{x^2-4}{x-2}$ at $x=2$: both num and den → 0; factor to get $x+2$ → removable.
• $\frac{x+1}{x-2}$ at $x=2$: num → 3, den → 0 → vertical asymptote.

Limits at Infinity (Horizontal Asymptotes)

Horizontal asymptote at $y = L$: $\lim_{x\to\infty}f(x) = L$ or $\lim_{x\to-\infty}f(x) = L$

Key fact: $\lim_{x\to\infty}\frac{1}{x^n} = 0$ for any $n > 0$

Rational functions — three cases:

Let $p$ = degree of numerator, $q$ = degree of denominator.

Case	Rule	Horizontal asymptote
$p < q$	$\lim = 0$	$y = 0$
$p = q$	$\lim = \frac{\text{leading coef}(p)}{\text{leading coef}(q)}$	$y = \frac{a_n}{b_m}$
$p > q$	$\lim = \pm\infty$	None (oblique instead)

Technique for equal degrees: divide num and denom by $x^q$:

$\lim_{x\to\infty}\frac{3x^2-7}{5x^2+2} = \lim_{x\to\infty}\frac{3-7/x^2}{5+2/x^2} = \frac{3-0}{5+0} = \frac{3}{5}$

Asymptotes of Other Functions

$e^x$: $\lim_{x\to-\infty}e^x = 0$ → horizontal asymptote $y = 0$ as $x\to-\infty$. No vertical asymptotes.

$\ln x$: vertical asymptote $x = 0$; no horizontal asymptote.

$\arctan x$: $\lim_{x\to\pm\infty}\arctan x = \pm\pi/2$ → horizontal asymptotes $y = \pm\pi/2$.