Limits At Infinity

Limits at infinity describe how a function behaves as $x$ approaches $\pm\infty$ . When $\lim_{x\to\infty} f(x) = L$ for a finite $L$ , the line $y = L$ is a horizontal asymptote. For rational functions, compare numerator and denominator degrees; for radicals, remember $\sqrt{x^2} = |x|$ to avoid sign errors.

Limits at Infinity and End Behavior

Limits at infinity describe what happens to $f(x)$ as $x$ grows without bound toward $+\infty$ or $-\infty$ : $\lim_{x\to\infty} f(x) = L \quad \text{or} \quad \lim_{x\to -\infty} f(x) = L$

This captures the end behavior of a function — how it behaves 'far out' on the graph. If either limit equals a finite number $L$ , then $y = L$ is a horizontal asymptote: the graph flattens and hugs this line as $x$ moves toward infinity. A function may have two different horizontal asymptotes (one on each side) or the same one on both sides, or none.

Quick Reference: Rational Functions

For $f(x) = \frac{p(x)}{q(x)}$ , the end behavior depends entirely on comparing degrees:

Relationship	Limit	Horizontal Asymptote
$\deg(p) < \deg(q)$	$0$	$y = 0$
$\deg(p) = \deg(q)$	$\frac{\text{leading coeff of }p}{\text{leading coeff of }q}$	Yes (ratio of coefficients)
$\deg(p) > \deg(q)$	$\pm\infty$	None

As $x \to \pm\infty$ , highest-degree terms dominate, so divide numerator and denominator by the highest power to isolate leading behavior.

Functions with Radicals: Watch Your Signs

For radicals, divide by $x^n$ matching the highest power under the root. Critically: $\sqrt{x^2} = |x|$ , not $x$ . This means:

For $x > 0$ : $\sqrt{x^2} = x$
For $x < 0$ : $\sqrt{x^2} = -x$

Consider $\lim_{x\to\infty} \frac{\sqrt{4x^2+1}}{x}$ . Factor: $\sqrt{4x^2+1} = x\sqrt{4+\frac{1}{x^2}} \to 2x$ as $x\to\infty$ , so the limit is $2$ . But as $x\to-\infty$ , $\sqrt{x^2} = -x$ , flipping the sign to give a limit of $-2$ .

This produces two different horizontal asymptotes: $y=2$ (right side) and $y=-2$ (left side) — a common source of errors when students forget the absolute value rule.