Limit Calculator
Calculate limits of functions as x approaches a value. Evaluate one-sided limits, limits at infinity, and indeterminate forms with step-by-step solutions.
Polynomial: ax² + bx + c
f(x) = x^2 - 4
Enter a value for x to approach
Step-by-Step Solution
Polynomial function
f(x) = x^2 - 4
Limit Rules
- • Polynomials: Always use direct substitution (continuous everywhere)
- • Rational: Check denominator first - may have discontinuity
- • 0/0: Indeterminate - factor, simplify, or use L'Hôpital's Rule
- • ∞/∞: Also indeterminate - compare degrees or use L'Hôpital's
- • c/0 (c≠0): Infinite limit - check sign from both sides
Related Calculators
About This Calculator
The Limit Calculator evaluates mathematical limits as a variable approaches a specific value or infinity. Whether you are finding one-sided limits, handling indeterminate forms like 0/0 or infinity/infinity, calculating limits at infinity, or applying L'Hopital's Rule, this calculator provides accurate results with step-by-step solutions. Limits are the foundation of calculus and essential for understanding continuity, derivatives, and integrals. Students often encounter limits when first studying calculus, and this calculator helps verify homework answers while showing the solution process. Our tool handles polynomial, rational, trigonometric, exponential, and logarithmic functions, automatically detecting and resolving indeterminate forms. It shows the limit from the left (x approaches from below), limit from the right (x approaches from above), and determines if the two-sided limit exists. Perfect for calculus students, teachers creating examples, engineers analyzing system behavior, and anyone working with mathematical analysis and continuous functions.
How to Use the Limit Calculator
- 1Enter your function f(x) using standard mathematical notation.
- 2Specify the value that x approaches (a number or infinity).
- 3Choose the limit direction: left, right, or both.
- 4Click Calculate to evaluate the limit.
- 5View the step-by-step solution showing the method used.
- 6For indeterminate forms, see how L'Hopital's Rule is applied.
- 7Check if the limit exists and its final value.
Limit Notation and Definition
A limit describes the value a function approaches as the input approaches some value.
Limit Notation:
lim (x -> a) f(x) = L
This means: As x gets arbitrarily close to a, f(x) gets arbitrarily close to L.
Types of Limits:
- Two-sided limit: x approaches a from both directions
- Left-hand limit: x approaches a from below (x -> a-)
- Right-hand limit: x approaches a from above (x -> a+)
- Limit at infinity: x approaches positive or negative infinity
A two-sided limit exists only if both one-sided limits exist and are equal.
Indeterminate Forms
Indeterminate forms require special techniques to evaluate:
| Form | Example | Method |
|---|---|---|
| 0/0 | lim (sin x)/x as x->0 | L'Hopital's Rule or algebraic manipulation |
| inf/inf | lim x^2/e^x as x->inf | L'Hopital's Rule |
| 0 * inf | lim x * ln(x) as x->0+ | Rewrite as fraction, then L'Hopital |
| inf - inf | lim (1/x - 1/sin x) | Combine fractions, then simplify |
| 1^inf | lim (1+1/x)^x as x->inf | Use logarithms |
| 0^0 | lim x^x as x->0+ | Use logarithms |
| inf^0 | lim x^(1/x) as x->inf | Use logarithms |
L'Hopital's Rule: If lim f(x)/g(x) gives 0/0 or inf/inf, then lim f(x)/g(x) = lim f'(x)/g'(x)
Common Limit Values
Important limits to memorize for calculus:
| Limit | Value |
|---|---|
| lim (sin x)/x as x->0 | 1 |
| lim (1-cos x)/x as x->0 | 0 |
| lim (tan x)/x as x->0 | 1 |
| lim (1+1/x)^x as x->inf | e (approximately 2.718) |
| lim (e^x - 1)/x as x->0 | 1 |
| lim ln(1+x)/x as x->0 | 1 |
| lim (a^x - 1)/x as x->0 | ln(a) |
| lim x^n/e^x as x->inf | 0 (for any n) |
These limits form the basis for many derivative formulas and Taylor series expansions.
Pro Tips
- 💡Always check both left and right limits to determine if a two-sided limit exists.
- 💡For indeterminate forms (0/0, inf/inf), try L'Hopital's Rule or algebraic simplification.
- 💡Polynomial limits as x approaches a finite value can be found by direct substitution.
- 💡For rational functions at infinity, compare the degrees of numerator and denominator.
- 💡Memorize key limits like lim(sin x/x) = 1 as x approaches 0.
Frequently Asked Questions
A limit describes what value a function approaches as the input gets close to some point, while the function value is the actual output at that point. A function may have a limit at a point where it is undefined (like a hole in the graph), or the limit may differ from the function value (like at a jump discontinuity).
