Enter a function, choose your variable, set the approach value, and select a direction (two-sided, left, or right) to evaluate its limit. The Limit Calculator substitutes, simplifies, and applies standard limit techniques to return the limit value along with supporting details like indeterminate form detection and L'Hôpital's rule hints.
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A limit describes the value a function approaches as its input gets closer and closer to a specific point, without necessarily reaching it. For example, lim(x→2) of (x²-4)/(x-2) equals 4, even though the function is undefined at x = 2. Limits are the foundation of calculus, underpinning derivatives and integrals.
A left-hand limit (x → a⁻) examines the function's behavior as x approaches 'a' from values smaller than 'a'. A right-hand limit (x → a⁺) looks at approach from values larger than 'a'. For a two-sided limit to exist, both one-sided limits must exist and be equal.
Indeterminate forms like 0/0, ∞/∞, or 0·∞ arise when direct substitution is not sufficient to evaluate a limit. Techniques such as factoring and simplifying, rationalizing the denominator, or L'Hôpital's rule (differentiating numerator and denominator separately) are used to resolve them.
For limits at infinity, type 'Infinity' or '-Infinity' as the approach value. The calculator evaluates behavior as the variable grows without bound. For rational functions, divide numerator and denominator by the highest power of x to determine the horizontal asymptote.
Common methods include direct substitution (simplest case), factoring to cancel common terms, rationalizing using conjugates, applying limit laws (sum, product, quotient rules), using known standard limits (e.g. sin(x)/x → 1 as x → 0), and L'Hôpital's rule for indeterminate forms.
Yes. A limit describes approach behavior, not the actual value at the point. For instance, f(x) = (x²−4)/(x−2) is undefined at x = 2 (division by zero), but its limit as x → 2 is 4 because the function simplifies to x + 2 for all x ≠ 2.
A limit does not exist when the left-hand and right-hand limits are not equal, when the function oscillates without settling on a value (e.g. sin(1/x) as x → 0), or when the function grows without bound toward ±infinity at the point of interest.
You can choose from common mathematical variables: x, t, u, v, or w. The default variable is x, which is standard for most single-variable calculus problems. Simply select your variable from the dropdown before entering your function.