Infinite Limits

Level 1 - Calculus topic page in Foundations And Limits.

Principle

An infinite limit describes values growing without bound as the input approaches a point. It signals vertical asymptotic behaviour rather than a finite output value.

Notation

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

f(x) grows without positive bound near a

\(\lim_{x\to a^-}\)

left-hand limit

\(\lim_{x\to a^+}\)

right-hand limit

The Core Method

For rational functions, infinite limits usually occur where the denominator approaches zero while the numerator approaches a non-zero value. Check the sign from each side.

Vertical asymptote signal

\[f(x)\to\pm\infty\text{ as }x\to a\]

The two one-sided limits may have opposite signs, so do not combine them until both sides have been checked.

Worked Cases

Question

Find the one-sided limits of \(f(x)=\frac{1}{x-2}\) as \(x\to2\).

Answer

As \(x\to2^+\), the denominator \(x-2\) is small and positive, so \(f(x)\to\infty\). As \(x\to2^-\), the denominator is small and negative, so \(f(x)\to-\infty\). The two-sided limit is not a single infinity with one sign.

Examples

Question

Find \(\lim_{x\to0}\frac{1}{x^2}\).

Answer

For \(x\ne0\), \(x^2\) is positive. As \(x\) approaches \(0\), \(x^2\) approaches \(0\) through positive values. Therefore \(1/x^2\to\infty\).

Continuity

Asymptotic Behaviour