Indeterminate form

An expression arising in a limit whose apparent form alone does not determine the limit’s value. Common forms include 0/0, ∞/∞, 0·∞, ∞−∞, 0⁰, 1^∞, and ∞⁰.

Limit of a function: The value a function approaches as its input approaches a specified point. An indeterminate form arises when evaluating a limit from the expressions of its parts.
One-sided limit: A limit taken as the input approaches a point from only one side. An apparent indeterminate form may have different one-sided behaviors near a discontinuity.
Limit comparison test: A series convergence test that compares positive terms through the limit of their ratio. The test relies on ratio limits, where ∞/∞ or 0/0 may need resolution.
Determinate form: An expression whose form alone fixes the result of the associated limit under the stated conditions. Unlike an indeterminate form, a determinate form already dictates the limit’s outcome.
L'Hôpital's rule: A theorem that evaluates certain limits of quotients by differentiating numerator and denominator. It resolves many 0/0 and ∞/∞ limits by replacing the original quotient with a derivative quotient.
Continuity: A property of a function whose value agrees with its limit at a point. Continuity can make direct substitution sufficient, bypassing indeterminate forms.
Derivative: The instantaneous rate of change of a function with respect to its input. L'Hôpital's rule turns certain indeterminate limits into limits involving derivatives.
Undefined expression: An expression that has no value under the rules of the arithmetic system being used. A quotient such as 0/0 is undefined at a point, but its surrounding limit may still exist.
Algebraic simplification: The rewriting of an expression into an equivalent, more useful form. Factoring or rationalizing can expose the value hidden by a 0/0 expression.
Removable discontinuity: A discontinuity caused by a missing or mismatched function value despite an existing finite limit. A 0/0 expression often signals a removable discontinuity after cancellation.

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