| Input | Result | |
|---|---|---|
| lim(x→2) (x²−4)/(x−2) | 4 | |
| lim(x→0) sin(x)/x | 1 | |
| lim(x→∞) (3x²+1)/(2x²) | 3/2 | |
| lim(x→0) (1−cos x)/x² | 1/2 |
How to Evaluate Limits
A limit describes what value a function approaches as its input approaches a target — regardless of whether the function is defined at that target. The techniques depend on the form of the expression.
Direct Substitution: If f is continuous at x=a, substitute directly. lim(x→3)(x²+2x) = 9+6 = 15.
Factoring: For 0/0 rational forms, factor and cancel. lim(x→2)(x²−4)/(x−2) = lim(x+2) = 4.
Rationalisation: For square root 0/0 forms, multiply by the conjugate. lim(x→9)(√x−3)/(x−9) = 1/(2√9) = 1/6.
L'Hôpital's Rule: For persistent 0/0 or ∞/∞ forms, differentiate numerator and denominator. lim(x→0)sin(x)/x: apply → cos(0)/1 = 1.
Standard trig limits: lim(x→0)sin(x)/x = 1. lim(x→0)(1−cos x)/x = 0. These are proved using the Squeeze Theorem.
How to Use This Limit Calculator
Enter your expression in the input box above using standard mathematical notation. Use ^ for exponents (e.g., x^3 for x³), * for multiplication when needed, sin(), cos(), tan(), ln(), sqrt() for standard functions. Then click Calculate to get your answer with full step-by-step working.
This calculator handles polynomial, trigonometric, exponential, logarithmic expressions, and combinations thereof. Results are shown in simplified form where possible, with each step of the working displayed below the answer.
For best results, enter expressions clearly without ambiguity. Use parentheses to group terms: (x^2 + 1)/(x - 1) rather than x^2+1/x-1. The calculator follows standard order of operations.
Understanding What Limits Actually Compute
A limit answers the question: as x approaches some value a, what value does f(x) get closer and closer to? The key word is approaches — the limit is about behavior near a point, not at the point itself. This distinction is what makes limits the foundation of calculus: we can describe instantaneous rates (derivatives) and infinite sums (integrals) by expressing them as limits of things we already understand.
Limits exist in three fundamental forms. Two-sided limits lim(x→a) f(x) require the function to approach the same value from both sides — if the left-hand limit and right-hand limit differ, the two-sided limit does not exist. One-sided limits lim(x→a⁺) or lim(x→a⁻) consider approach from only one direction — these always exist for "reasonable" functions even when the two-sided limit fails. Limits at infinity lim(x→∞) f(x) describe the long-run behavior and determine horizontal asymptotes.
The Algebra of Limits
Limits obey algebraic rules that make computation systematic. Sum rule: lim[f+g] = lim f + lim g. Product rule: lim[f·g] = (lim f)·(lim g). Quotient rule: lim[f/g] = (lim f)/(lim g) provided lim g ≠ 0. Power rule: lim[f]ⁿ = (lim f)ⁿ. These rules mean that for continuous functions (polynomials, trig, exponential), you can simply substitute the target value directly — direct substitution works whenever the result is a defined number. The challenge only arises in indeterminate forms like 0/0, ∞/∞, 0·∞, ∞−∞, 1^∞, 0⁰, and ∞⁰.
The Epsilon-Delta Definition
The formal definition: lim(x→a) f(x) = L means that for every ε > 0 (no matter how small a tolerance you specify for the output), there exists δ > 0 (a corresponding tolerance for the input) such that whenever 0 < |x−a| < δ, we have |f(x)−L| < ε. In plain English: you can make f(x) as close to L as anyone demands, by keeping x sufficiently close to a. This definition makes "approaching" rigorous and eliminates ambiguity from all of calculus.
Continuity and Its Relationship to Limits
A function is continuous at x = a if and only if three conditions hold: f(a) is defined, lim(x→a) f(x) exists, and these two values are equal. Continuity is the property that makes direct substitution valid — for continuous functions, evaluating a limit is as simple as plugging in the value. The major continuous function families (polynomials, rational functions on their domains, sin, cos, eˣ, ln x) can all be evaluated by direct substitution wherever they're defined.
- Stewart, J. (2015). Calculus, Ch. 2. Cengage.
- Spivak, M. (2006). Calculus, Ch. 5. Publish or Perish.
- Apostol, T. (1967). Calculus, Vol. 1. Wiley.