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Indeterminate Form

Definition

An indeterminate form is a symbolic expression involving 0,0\text{,} 1,1\text{,} and ∞\infty that results from applying limit laws to a limit expression, and which fails to uniquely determine the value or divergence of the limit. There are mainly seven typical indeterminate forms:

00, ∞∞, 0×∞, ∞−∞,00, 1∞, and ∞0.\frac{0}{0},\,\frac{\infty}{\infty},\,0\times \infty,\,\infty-\infty,\qquad 0^0,\,1^\infty,\text{ and } \infty^0.

Interpretation

An indeterminate form is a pattern of limits, such as a numerator and a denominator that both tend to 00, which the limit laws cannot settle. The word describes the form, not the limit: knowing only that f→0f\to0 and g→0g\to0 does not tell what f/gf/g does, so further analysis is required, and the limit may well exist.

The same form can lead to any outcome. As x→0+x\to0^{+}, the quotients x/xx/x, x2/xx^2/x and x/x2x/x^2 all have the form 0/00/0, but tend to 11, 00 and +∞+\infty. Likewise, (x+1)−x→1(x+1)-x\to1 and x2−x→+∞x^2-x\to+\infty as x→+∞x\to+\infty, although both have the form ∞−∞\infty-\infty. By contrast, forms such as 0/∞0/\infty, ∞+∞\infty+\infty and 0+∞0^{+\infty} are not indeterminate: they always give 00, +∞+\infty and 00.

The symbols +∞+\infty and −∞-\infty describe divergent quantities, including those arising in limits at infinity; they are not real numbers on which ordinary arithmetic determines the answer. For example, two divergent terms can have a finite, infinite or oscillating difference.

The usual tools are algebra (factorizing, cancelling, multiplying by a conjugate), asymptotic equivalence and L'Hôpital's rule, which handles 0/00/0 and ∞/∞\infty/\infty. The other forms are first rewritten: f⋅g=f/(1/g)f\cdot g=f/(1/g) turns 0×∞0\times\infty into a quotient, and fg=egln⁡ff^g=\mathrm{e}^{g\ln f} turns 000^0, 1∞1^\infty and ∞0\infty^0 into the form 0×∞0\times\infty in the exponent.

A common slip is to read 1∞1^\infty as 11. The base tends to 11 while the exponent grows, and the two effects compete: (1+1n)n→e(1+\tfrac{1}{n})^n\to\mathrm{e}, which is how Euler's number is defined.

Key Results

  • An indeterminate form does not determine the value of a limit.
  • Different expressions with the same symbolic form may have different limits.
  • Resolving indeterminate forms often requires algebraic manipulation.

Examples

  • lim⁡x→0sin⁡xx\displaystyle \lim_{x\to 0}\frac{\sin x}{x} has the form "00\dfrac{0}{0}" and it converges to 1.1\text{.}
  • lim⁡x→+∞e−xx\lim\limits_{x\to +\infty} \mathrm{e}^{-x} x has the form "0×∞0\times \infty" and it converges to 0.0\text{.}

Prerequisites

Sources

Symbols: infinity

Last updated: 22 October 2026 19:04