Infinite Limits

By SkytrailGroup Mathematics Editorial Team April 12, 2026

Definition

\(\lim_{{x\to\infty}}f(x)=L\) if f(x)→L as x grows without bound. Horizontal asymptote at y=L.

Examples

Example 1. \(\lim_{{x\to\infty}}\frac{{1}}{{x}}\)

Solution. = 0.

In Depth

Infinite limits describe the behavior of functions as the input grows without bound (limits at infinity) or as the function value grows without bound (infinite limits at a point). Both are essential for understanding asymptotic behavior and vertical/horizontal asymptotes.

Horizontal asymptotes: \(\lim_{x\to\infty} f(x) = L\) means the graph approaches the horizontal line \(y=L\). For rational functions, compare degrees: if degree of numerator < denominator, limit is 0; if equal, limit is the ratio of leading coefficients; if numerator degree is greater, the limit is \(\pm\infty\).

Vertical asymptotes occur where \(\lim_{x\to a} f(x) = \pm\infty\). For rational functions, these occur at zeros of the denominator that are not canceled by the numerator. The sign of the limit (\(+\infty\) or \(-\infty\)) depends on the sign of the function near \(a\).

L'Hôpital's Rule handles indeterminate forms \(0/0\) and \(\infty/\infty\) at infinity: \(\lim_{x\to\infty} f(x)/g(x) = \lim_{x\to\infty} f'(x)/g'(x)\). Other indeterminate forms (\(0\cdot\infty\), \(\infty-\infty\), \(0^0\), \(1^\infty\), \(\infty^0\)) are converted to \(0/0\) or \(\infty/\infty\) by algebraic manipulation.

Key Properties & Applications

Infinite limits describe vertical asymptotes: \(\lim_{x\to a}f(x)=\infty\) means \(f(x)\) grows without bound as \(x\to a\). Limits at infinity describe horizontal asymptotes: \(\lim_{x\to\infty}f(x)=L\) means \(f(x)\to L\). Rational functions have horizontal asymptotes determined by the ratio of leading coefficients.

L'Hôpital's rule resolves indeterminate forms \(0/0\) and \(\infty/\infty\): if \(\lim f/g\) is indeterminate, then \(\lim f/g=\lim f'/g'\) (under regularity conditions). Other indeterminate forms (\(0\cdot\infty\), \(\infty-\infty\), \(0^0\), \(1^\infty\), \(\infty^0\)) are reduced to \(0/0\) or \(\infty/\infty\) by algebraic manipulation.

Infinite limits are essential in asymptotic analysis: big-O notation, little-o notation, and asymptotic equivalence (\(f\sim g\) means \(f/g\to1\)) describe the growth rates of functions and algorithms. The prime number theorem \(\pi(x)\sim x/\ln x\) is an asymptotic statement.