Evaluate polynomial ratio limits using direct substitution and L'Hôpital's rule for indeterminate forms.
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Numerator terms
Denominator terms
Result
Step-by-step
Calculus
A limit calculator evaluates the limit of a rational function (polynomial divided by polynomial) as x approaches a given value. It uses direct substitution first, then applies L'Hôpital's rule for indeterminate forms like 0/0.
Start with direct substitution — plug the approach value into the function. If the result is a defined number, that is the limit.
If substitution yields 0/0 (indeterminate form), apply L'Hôpital's rule: take the derivative of the numerator and denominator separately, then re-evaluate the limit.
A worked example helps translate the limit calculator maths into a realistic scenario so the user can compare the headline result with a concrete set of inputs.
That matters because a result is easier to trust when the page shows how the same logic behaves in a practical case instead of leaving the formula abstract.
Use the limit calculator output as a planning aid, then compare it with the assumptions, units, and caveats shown elsewhere on the page before acting on the number alone.
That extra interpretation step matters because a calculator can simplify the arithmetic but still cannot replace real-world context such as local rules, contract terms, or individual circumstances.
Frequently asked questions
An indeterminate form like 0/0 or ∞/∞ means the limit cannot be determined by direct substitution alone. Additional techniques like L'Hôpital's rule or algebraic manipulation are needed.
When the numerator is nonzero and the denominator approaches zero, the limit is ±∞ (or does not exist), depending on the sign of the expressions near the approach value.
The safest manual check is to follow the same formula or rule one step at a time and compare that working with the calculator output. That catches sign errors, bracket mistakes, and input-order mixups without requiring any extra method beyond the underlying maths itself.
Guides
Step-by-step guides that use this calculator to solve real problems.
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