Concept explainerCalculus IFoundational8 min read

Infinite limits and vertical asymptotes

Infinity describes unbounded behavior, not a number a function eventually reaches.

\lim_{x\to a^+}f(x)=\infty

Concept explainer

What the idea means, why its conditions matter, and where it connects.

Reviewed July 11, 2026

MeaningStart here

An infinite limit says outputs can be made arbitrarily large in magnitude by taking inputs sufficiently close to the target from the stated side.

Infinity is behavior

The notation does not claim the limit exists as a real number. It records that the function grows without bound. That distinction matters when applying limit laws and interpreting graphs.

One-sided notation is essential because the two sides of a vertical asymptote may head in opposite directions.

Finding the sign on each side

Factor the denominator and use a sign chart near the suspect point. A tiny positive denominator and a positive numerator produce positive infinity; a tiny negative denominator reverses the direction.

Do not rely on a calculator snapshot, which can hide behavior behind scale or sampling.

Vertical versus horizontal asymptotes

Vertical asymptotes come from inputs approaching a finite boundary. Horizontal asymptotes describe outputs as inputs head to positive or negative infinity. A function may cross a horizontal asymptote; the statement concerns end behavior.

Worked exampleRead both sides

Analyze 1/(x − 2) near x = 2.

x\to2^-\Longrightarrow x-2<0

The denominator is a tiny negative number.

\lim_{x\to2^-}\frac1{x-2}=-\infty

The left side decreases without bound.

\lim_{x\to2^+}\frac1{x-2}=+\infty

The right side increases without bound.

Result

\boxed{x=2\text{ is a vertical asymptote}}

Watch for

Common mistakes

Treating infinity as a number that can be substituted.
Reporting a two-sided infinite limit when the sides disagree.
Assuming every denominator zero creates an asymptote without checking cancellation.

Keep

Three takeaways

Use one-sided limits near vertical asymptotes.
Sign analysis determines positive or negative infinity.
Check for removable cancellation first.