What does \lim_{x \to \infty} actually mean?

It asks what value the function approaches (or whether it grows without bound) as the input x becomes arbitrarily large and positive. The arrow means "approaches," so this reads as "the limit as x approaches infinity."

Do I always have to look at leading coefficients?

No—first compare the degrees n and m. Only if they are equal do you need to examine the leading coefficients; otherwise, the degree comparison alone tells you whether the limit is 0, \infty, or -\infty.

Can the limit be a negative number other than zero?

Yes. For example, \lim_{x \to \infty} \frac{2x}{3x} = \frac{2}{3} is positive, while \lim_{x \to \infty} \frac{-2x^2}{3x^2} = -\frac{2}{3} is negative. The sign depends on whether the leading coefficients a_n and b_m have the same sign or opposite signs.

What is a horizontal asymptote?

A horizontal asymptote is a horizontal line that the graph approaches as x \to \infty or x \to -\infty. If \lim_{x \to \infty} f(x) = L where L is a finite number, then the line y = L is a horizontal asymptote of the graph.

Limits at Infinity

Find what a rational function approaches as x grows indefinitely by comparing the degrees and leading coefficients of the numerator and denominator.

\lim_{x \to \infty} \frac{a_n x^n}{b_m x^m}

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What each symbol means

What Limits at Infinity takes

Symbol	Meaning
$a_n$	The leading coefficient (constant multiplier) of the numerator's highest-degree term; its sign determines whether the limit points toward positive or negative infinity.
$b_m$	The leading coefficient of the denominator's highest-degree term; it determines the limit's magnitude when the numerator and denominator degrees are equal.
$n$	The degree (the highest exponent) of the numerator polynomial; must be a non-negative integer and controls how fast the numerator grows as $x \to \infty$ .
$m$	The degree of the denominator polynomial; must be a non-negative integer and determines whether the denominator or numerator dominates the behavior as $x \to \infty$ .
$x$	The independent variable approaching infinity, representing how the function behaves for increasingly large positive values.

When to use it

Use this when you need to determine the horizontal asymptote or end behavior of a rational function as $x$ becomes very large.

Level

Usually taught in: Calculus I

Worked examples

1. Finding a limit when denominator degree exceeds numerator degree

Problem

Find

\lim_{x \to \infty} \frac{3x}{5x^2 + 2}

$\lim_{x \to \infty} \frac{3x}{5x^2 + 2}$
Set up the limit as $x$ approaches infinity.
$\approx \lim_{x \to \infty} \frac{3x}{5x^2}$
For very large $x$ , only the highest-degree terms matter: the numerator is degree 1 and the denominator is degree 2, so the constant 2 becomes negligible.
$\lim_{x \to \infty} \frac{3x}{5x^2} = \lim_{x \to \infty} \frac{3}{5x}$
Divide both numerator and denominator by $x$ to simplify.
$\lim_{x \to \infty} \frac{3}{5x} = 0$
As $x \to \infty$ , the denominator $5x$ becomes arbitrarily large while the numerator stays at 3, so the fraction shrinks to zero.

Answer: $0$

The denominator's degree (2) exceeds the numerator's degree (1), so the denominator grows much faster and dominates, driving the fraction toward zero.

2. Finding a limit when numerator degree exceeds denominator degree

Problem

Find

\lim_{x \to \infty} \frac{-3x^3 + 5x}{2x^2 - 7}

$\lim_{x \to \infty} \frac{-3x^3 + 5x}{2x^2 - 7}$
Set up the limit as $x$ approaches infinity.
$\approx \lim_{x \to \infty} \frac{-3x^3}{2x^2}$
For very large $x$ , focus on the highest-degree terms in the numerator ( $-3x^3$ ) and denominator ( $2x^2$ ).
$\lim_{x \to \infty} \frac{-3x^3}{2x^2} = \lim_{x \to \infty} \frac{-3x}{2}$
Simplify the powers: $\frac{x^3}{x^2} = x$ , so divide both numerator and denominator by $x^2$ .
$\lim_{x \to \infty} \frac{-3x}{2} = -\infty$
As $x \to \infty$ , the numerator $-3x$ becomes increasingly negative while the denominator stays at 2, so the fraction approaches $-\infty$ .

Answer: $-\infty$

The numerator's degree (3) exceeds the denominator's degree (2), so the numerator grows faster and dominates. The leading coefficient $-3$ is negative, which determines that the result is negative infinity.

3. Finding average speed on a two-leg road trip as distance increases

Problem

On a road trip, you drive 120 miles at 60 mph, then

x

additional miles at 75 mph. Your average speed for the entire trip is

\frac{120 + x}{2 + \frac{x}{75}}

. Find the limit of the average speed as

x \to \infty

$\lim_{x \to \infty} \frac{120 + x}{2 + \frac{x}{75}}$
Set up the limit representing the average speed as the second leg's distance $x$ approaches infinity.
$\lim_{x \to \infty} \frac{120 + x}{2 + \frac{x}{75}} = \lim_{x \to \infty} \frac{75(120 + x)}{75(2 + \frac{x}{75})} = \lim_{x \to \infty} \frac{9000 + 75x}{150 + x}$
Multiply both numerator and denominator by 75 to clear the fraction in the denominator.
$n = 1 \text{ (numerator)}, \quad m = 1 \text{ (denominator)}, \quad a_1 = 75, \quad b_1 = 1$
Both polynomials now have degree 1, with leading coefficients 75 in the numerator and 1 in the denominator.
$\lim_{x \to \infty} \frac{9000 + 75x}{150 + x} = \frac{75}{1} = 75$
When degrees are equal, the limit equals the ratio of the leading coefficients.

Answer: $75 \text{ mph}$

The trip begins with 2 hours at 60 mph, but this fixed time becomes negligible compared to the total time as the second leg grows longer. Eventually, your overall average speed is determined by the speed on the longer portion of the trip, which is 75 mph.

Common mistakes

Where Limits at Infinity usually goes wrong

The mistake	Why it is wrong	The fix
Writing $\lim_{x \to \infty} \frac{3x^2 + 2x}{5x^2 + 1} = 3 + 2 = 5$	This incorrectly adds all coefficients together instead of comparing the degrees first and focusing only on the leading coefficients when degrees are equal.	When degrees are equal, divide only the leading coefficients: $\frac{3}{5}$ , ignoring the other terms.
Assuming $\lim_{x \to \infty} \frac{2x^3}{x^2} = 0$ because "larger functions go to zero at infinity"	This reverses the correct principle: when the numerator's degree exceeds the denominator's degree, the function grows without bound (toward $\pm\infty$ ), not toward zero.	Compare degrees first: if $n > m$ , the limit is $\infty$ or $-\infty$ depending on the sign of $a_n$ ; if $n < m$ , the limit is zero.
Confusing degrees and writing $\lim_{x \to \infty} \frac{2x^2 + 3}{5x^2 + 1} = \infty$ instead of $\frac{2}{5}$	When the numerator and denominator degrees are equal, the limit is a finite number (the ratio of leading coefficients), not infinity.	Remember the three cases: degrees equal $\Rightarrow$ finite limit; numerator degree larger $\Rightarrow$ infinity limit; denominator degree larger $\Rightarrow$ zero limit.

Tips and when to use something else

Degree is everything: compare $n$ and $m$ first before performing any arithmetic. The relationship between these numbers determines the entire answer.
If degrees are equal, you only need the leading coefficients $a_n$ and $b_m$ ; all other terms vanish as $x \to \infty$ .
For polynomial limits at infinity, you can also use L'Hopital's Rule, but degree comparison is usually faster and does not require calculus.
Watch the sign of the leading coefficient: if $n > m$ and $a_n$ is negative, the limit is $-\infty$ , not $+\infty$ .

Related formulas

Limit of a Rational FunctionFind the limit of a rational function at a point by substituting the point directly, provided the denominator is nonzero.Asymptotes from LimitsUse limits at infinity to determine where a function is heading as x grows without bound, then identify that destination as a horizontal asymptote equation.Indeterminate FormsIndeterminate forms are expressions like 0/0 and ∞/∞ that arise when evaluating limits, signaling that direct substitution doesn't work.L'Hopital's RuleL'Hopital's Rule efficiently solves limits that reduce to 0/0 or ∞/∞ forms by taking derivatives of the numerator and denominator separately.

Frequently asked questions

What does $\lim_{x \to \infty}$ actually mean?: It asks what value the function approaches (or whether it grows without bound) as the input $x$ becomes arbitrarily large and positive. The arrow means "approaches," so this reads as "the limit as $x$ approaches infinity."
Do I always have to look at leading coefficients?: No—first compare the degrees $n$ and $m$ . Only if they are equal do you need to examine the leading coefficients; otherwise, the degree comparison alone tells you whether the limit is 0, $\infty$ , or $-\infty$ .
Can the limit be a negative number other than zero?: Yes. For example, $\lim_{x \to \infty} \frac{2x}{3x} = \frac{2}{3}$ is positive, while $\lim_{x \to \infty} \frac{-2x^2}{3x^2} = -\frac{2}{3}$ is negative. The sign depends on whether the leading coefficients $a_n$ and $b_m$ have the same sign or opposite signs.
What is a horizontal asymptote?: A horizontal asymptote is a horizontal line that the graph approaches as $x \to \infty$ or $x \to -\infty$ . If $\lim_{x \to \infty} f(x) = L$ where $L$ is a finite number, then the line $y = L$ is a horizontal asymptote of the graph.

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Reviewed 2026-09-18