If the limit exists, does that guarantee the graph never touches the asymptote?

No. A horizontal asymptote describes only what happens as x \to \infty (or x \to -\infty), not the behaviour everywhere. The graph can cross the line y = L at any finite x and still satisfy \lim_{x \to \infty} f(x) = L. The function must get arbitrarily close to L for very large x, but it does not have to stay on one side of the line to get there.

What happens if the limit is \infty or -\infty?

If \lim_{x \to \infty} f(x) = \infty or \lim_{x \to \infty} f(x) = -\infty, then there is no horizontal asymptote. The function grows (or shrinks) without bound instead of approaching a finite value. You can use Infinite Limits to describe this behavior more precisely.

Why should I check the limit as x \to -\infty as well?

A function can behave very differently as x \to \infty and as x \to -\infty. For example, f(x) = \frac{x}{|x|} approaches 1 from the right but approaches -1 from the left, giving two different horizontal asymptotes. A complete analysis requires checking both directions, even though this rule statement only mentions x \to \infty.

How do I tell the difference between a horizontal asymptote and a slant (oblique) asymptote?

A horizontal asymptote occurs when \lim_{x \to \infty} f(x) equals a finite constant—the graph approaches a horizontal line. A slant asymptote occurs when the function approaches a non-horizontal line, typically when the numerator's polynomial degree is exactly one more than the denominator's. You can find slant asymptotes using polynomial long division or L'Hopital's Rule.

Asymptotes from Limits

Use limits at infinity to determine where a function is heading as x grows without bound, then identify that destination as a horizontal asymptote equation.

\lim_{x \to \infty} f(x) = L \implies y = L \text{ is a horizontal asymptote}

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What each symbol means

What Asymptotes from Limits takes

Symbol	Meaning
$f$	A function whose behavior you are studying; could be anything from a simple rational function to a more complex expression. If you treat $f$ as a number instead of a function, you cannot apply this rule.
$x$	The independent variable, typically representing input or position. The statement looks at what happens as $x$ becomes arbitrarily large (grows toward positive infinity), not at any single value.
$L$	A finite real number that $f(x)$ approaches as $x \to \infty$ ; this is the limit value. If $L$ is infinite or does not exist, this rule does not give you a horizontal asymptote.
$y$	The dependent variable or output of the function; in the equation $y = L$ , it represents the horizontal line itself. This line is where the graph gets closer and closer as $x$ increases without bound.

When to use it

When you need to describe the long-term behavior of a function or find where its graph approaches a horizontal line.

Level

Usually taught in: Calculus I

Worked examples

1. Simple rational function with zero asymptote

Problem

Find

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

and identify the horizontal asymptote.

$A(x) = \frac{5}{x + 2}$
We start with the function and need to find where it approaches as $x$ becomes very large.
$= \frac{5}{x(1 + \frac{2}{x})} = \frac{5}{x} \cdot \frac{1}{1 + \frac{2}{x}}$
We factor $x$ from the denominator so we can see what each piece does as $x \to \infty$ .
$\lim_{x \to \infty} \left( \frac{5}{x} \cdot \frac{1}{1 + \frac{2}{x}} \right) = 0 \cdot \frac{1}{1 + 0} = 0$
As $x \to \infty$ , we have $\frac{5}{x} \to 0$ and $\frac{2}{x} \to 0$ , so the product approaches $0 \cdot 1 = 0$ .

Answer: $y = 0$

The function approaches 0 as $x$ grows without bound, so $y = 0$ is a horizontal asymptote. The graph will get arbitrarily close to the $x$ -axis for large positive values of $x$ , but will never reach it.

2. Rational function with non-zero, non-integer asymptote

Problem

Find

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

and identify the horizontal asymptote.

$\lim_{x \to \infty} \frac{3x - 2}{2x + 5}$
For rational functions where the numerator and denominator have the same degree, divide both by the highest power of $x$ .
$= \lim_{x \to \infty} \frac{x(3 - \frac{2}{x})}{x(2 + \frac{5}{x})} = \lim_{x \to \infty} \frac{3 - \frac{2}{x}}{2 + \frac{5}{x}}$
We factor $x$ from both numerator and denominator, then cancel the common factor.
$= \frac{3 - 0}{2 + 0} = \frac{3}{2}$
As $x \to \infty$ , both $\frac{2}{x} \to 0$ and $\frac{5}{x} \to 0$ , leaving the ratio of leading coefficients.

Answer: $y = \frac{3}{2}$

For rational functions with equal-degree polynomials, the horizontal asymptote is always the ratio of the leading coefficients. Here the limit is $\frac{3}{2}$ , so the graph approaches the line $y = \frac{3}{2}$ as $x$ increases.

3. Phone plan: average cost per gigabyte approaches a constant

Problem

A wireless provider charges a $60 monthly service fee plus $3 per gigabyte of data used. The average monthly cost per gigabyte is

A(x) = \frac{60 + 3x}{x}

dollars, where

x

is gigabytes used. Find

\lim_{x \to \infty} A(x)

and identify the horizontal asymptote.

$A(x) = \frac{60 + 3x}{x}$
This represents the total monthly bill divided by the number of gigabytes—the average cost per gigabyte.
$= \frac{60}{x} + \frac{3x}{x} = \frac{60}{x} + 3$
We split the single fraction into two pieces by dividing each term of the numerator by the denominator.
$\lim_{x \to \infty} A(x) = \lim_{x \to \infty} \left( \frac{60}{x} + 3 \right) = 0 + 3 = 3$
As $x \to \infty$ , the fraction $\frac{60}{x} \to 0$ because a fixed numerator divided by a growing denominator shrinks toward 0.

Answer: $y = 3$

As the customer uses more and more data, the fixed $60 service fee gets spread across an increasing number of gigabytes, making its contribution to the average cost negligible. The average cost per gigabyte therefore approaches the per-gigabyte rate of$ 3. This means the graph has a horizontal asymptote at $y = 3$ .

Common mistakes

Where Asymptotes from Limits usually goes wrong

The mistake	Why it is wrong	The fix
Looking at the graph of $f(x) = \frac{x - 1}{x}$ , observing that it seems to approach a horizontal line, and claiming without calculation that $y = 0$ is a horizontal asymptote.	Visual inspection of a graph is not a rigorous proof. You must verify the asymptote by computing the limit algebraically using the definition.	Compute $\lim_{x \to \infty} \frac{x - 1}{x} = \lim_{x \to \infty} \left(1 - \frac{1}{x}\right) = 1$ , so the actual horizontal asymptote is $y = 1$ , not $y = 0$ .
Computing $\lim_{x \to \infty} f(x) = L$ correctly but then writing ' $y = L$ is a vertical asymptote' or confusing it with a different type of asymptote.	Horizontal asymptotes and vertical asymptotes are fundamentally different. A horizontal asymptote comes from a limit at infinity; a vertical asymptote comes from a limit approaching a finite value where the function is undefined.	Use correct terminology: if $\lim_{x \to \infty} f(x) = L$ , then $y = L$ is a horizontal asymptote (a horizontal line), not a vertical one.
Computing $\lim_{x \to \infty} (x^2 + 1) = \infty$ and then writing ' $y = \infty$ is an asymptote' or concluding the function has an asymptote.	Infinity is not a real number, so a 'horizontal asymptote at $y = \infty$ ' does not exist. A horizontal asymptote must be a finite constant, and the rule only applies when $L$ is a real number.	If $\lim_{x \to \infty} f(x) = \infty$ or if the limit does not exist, then there is no horizontal asymptote from this rule. The function grows without bound instead.

Tips and when to use something else

For rational functions, divide both numerator and denominator by the highest power of $x$ that appears—this is the quickest way to find horizontal asymptotes.
Always check $\lim_{x \to -\infty}$ separately if you need a complete picture. A function can have two different horizontal asymptotes (one as $x \to \infty$ and another as $x \to -\infty$ ), or an asymptote in one direction but not the other. Use Limits at Infinity for both directions.
Graphs are allowed to cross their horizontal asymptotes at finite $x$ values. A horizontal asymptote describes only the long-term behavior as $x$ grows arbitrarily large, not what happens everywhere.
If the limit at infinity does not exist (for example, if the function oscillates forever), there is no horizontal asymptote. Use Infinite Limits or graphing to understand the function's end behavior in that case.

Related formulas

Limits at InfinityFind what a rational function approaches as x grows indefinitely by comparing the degrees and leading coefficients of the numerator and denominator.Limit of a Rational FunctionFind the limit of a rational function at a point by substituting the point directly, provided the denominator is nonzero.Infinite LimitsInfinite Limits describe when a function grows unboundedly as the input approaches a specific value, letting you identify and analyze vertical asymptotes.Definition of a LimitDefinition of a Limit describes what value a function approaches as its input approaches a specific point—essential for all of calculus.

Frequently asked questions

If the limit exists, does that guarantee the graph never touches the asymptote?: No. A horizontal asymptote describes only what happens as $x \to \infty$ (or $x \to -\infty$ ), not the behaviour everywhere. The graph can cross the line $y = L$ at any finite $x$ and still satisfy $\lim_{x \to \infty} f(x) = L$ . The function must get arbitrarily close to $L$ for very large $x$ , but it does not have to stay on one side of the line to get there.
What happens if the limit is $\infty$ or $-\infty$ ?: If $\lim_{x \to \infty} f(x) = \infty$ or $\lim_{x \to \infty} f(x) = -\infty$ , then there is no horizontal asymptote. The function grows (or shrinks) without bound instead of approaching a finite value. You can use Infinite Limits to describe this behavior more precisely.
Why should I check the limit as $x \to -\infty$ as well?: A function can behave very differently as $x \to \infty$ and as $x \to -\infty$ . For example, $f(x) = \frac{x}{|x|}$ approaches $1$ from the right but approaches $-1$ from the left, giving two different horizontal asymptotes. A complete analysis requires checking both directions, even though this rule statement only mentions $x \to \infty$ .
How do I tell the difference between a horizontal asymptote and a slant (oblique) asymptote?: A horizontal asymptote occurs when $\lim_{x \to \infty} f(x)$ equals a finite constant—the graph approaches a horizontal line. A slant asymptote occurs when the function approaches a non-horizontal line, typically when the numerator's polynomial degree is exactly one more than the denominator's. You can find slant asymptotes using polynomial long division or L'Hopital's Rule.

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Reviewed 2026-09-18