Free Limit Calculator with Steps
Evaluate limits step-by-step. Learn direct substitution, L'Hôpital's rule, and special limit techniques with instant feedback. Perfect for A-Level and university calculus.
Step-by-step learning with explanations
Enter an Expression
Enter a mathematical expression above to learn how to evaluate the limit step-by-step.
What is a Limit?
A limit describes the value a function approaches as the input approaches a particular value. Written as lim(x→a) f(x) = L, it means "as x gets closer and closer to a, f(x) gets closer and closer to L".
Limits are fundamental to calculus. They're used to define derivatives (rate of change) and integrals (area under a curve), making them the foundation of all calculus concepts.
Limit Notation
How to Evaluate Limits
Direct Substitution
Try plugging in the value directly. If you get a real number (not 0/0 or ∞/∞), that's your answer!
Factoring & Simplifying
For 0/0 forms, factor the numerator and denominator, cancel common terms, then substitute again.
L'Hôpital's Rule
For 0/0 or ∞/∞ forms, differentiate top and bottom separately, then re-evaluate. Repeat if needed.
Special Limits
Memorise key limits: sin(x)/x → 1, (1-cos(x))/x → 0, (eˣ-1)/x → 1 as x → 0.
Indeterminate Forms
These forms require special techniques like L'Hôpital's rule or algebraic manipulation.
Frequently Asked Questions
How do I evaluate a limit?
Start with direct substitution. If indeterminate, try factoring, L'Hôpital's rule, or algebraic manipulation.
What does "limit does not exist" mean?
The function approaches different values from left and right, or oscillates, or diverges in different directions.
When do I use L'Hôpital's rule?
When direct substitution gives 0/0 or ∞/∞. Differentiate numerator and denominator separately.
What is lim(x→0) sin(x)/x?
This famous limit equals 1. It's fundamental to calculus and used in proving derivative formulas.
What are one-sided limits?
Left limit (x→a⁻) approaches from below; right limit (x→a⁺) approaches from above. Both must match for the limit to exist.
Is this suitable for A-Level revision?
Yes! Covers all A-Level techniques: direct substitution, factoring, L'Hôpital's rule, and special limits.
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