Online Limit Calculator - Wolfram|Alpha

A handy tool for solving limit problems

Wolfram|Alpha computes both one-dimensional and multivariate limits with great ease. Determine the limiting values of various functions, and explore the visualizations of functions at their limit points with Wolfram|Alpha.

Learn more about:

• One-dimensional limits »
• Multivariate limits »

Tips for entering queries

Use plain English or common mathematical syntax to enter your queries. For specifying a limit argument x and point of approach a, type "x -> a". For a directional limit, use either the + or – sign, or plain English, such as "left," "above," "right" or "below."

• limit sin(x)/x as x -> 0
• limit (1 + 1/n)^n as n -> infinity
• lim ((x + h)^5 - x^5)/h as h -> 0
• lim (x^2 + 2x + 3)/(x^2 - 2x - 3) as x -> 3

• lim x/|x| as x -> 0
• limit tan(t) as t -> pi/2 from the left
• limit xy/(Abs(x) + Abs(y)) as (x,y) -> (0,0)
• limit x^2y^2/(x^4 + 5y^5) as (x,y) -> (0,0)

• View more examples »

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• Step-by-step solutions »

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What are limits?

Limits, a foundational tool in calculus, are used to determine whether a function or sequence approaches a fixed value as its argument or index approaches a given point.

Limits can be defined for discrete sequences, functions of one or more real-valued arguments or complex-valued functions. For a sequence { Start subscript, Start base, x , base End,Start subscript, n , subscript End , subscript End }xn indexed on the natural number set n ∈ Nn∈N, the limit LL is said to exist if, as n → ∞n→∞, the value of the elements of { Start subscript, Start base, x , base End,Start subscript, n , subscript End , subscript End }xn get arbitrarily close to LL.

A real-valued function f (x)fx is said to have a limit LL if, as its argument xx is taken arbitrarily close to Start subscript, Start base, x , base End,Start subscript, 0 , subscript End , subscript Endx0, its value can be made arbitrarily close to LL. Formally defined, a function f (x)fx has a finite limit L = Start Limit, Start variable, x , variable End,Start target value, Start subscript, Start base, x , base End,Start subscript, 0 , subscript End , subscript End , target value End,Start expression, f (x) , expression End , Limit EndL=limxmm-template-arrow-right-8x0fx at point Start subscript, Start base, x , base End,Start subscript, 0 , subscript End , subscript Endx0 if, for all ε > 0ε>0 , there exists δ > 0δ>0 such that Start Absolute, Start argument, f (x) - L , argument End , Absolute End < εfx-L<ε whenever Start Absolute, Start argument, x - Start subscript, Start base, x , base End,Start subscript, 0 , subscript End , subscript End , argument End , Absolute End < δx-x0<δ. This definition can be further extended for LL or Start subscript, Start base, x , base End,Start subscript, 0 , subscript End , subscript Endx0 being taken to infinity and to multivariate and complex functions.

For functions of one real-valued variable, the limit point Start subscript, Start base, x , base End,Start subscript, 0 , subscript End , subscript Endx0 can be approached from either the right/above (denoted Start Limit from the Right, Start variable, x , variable End,Start target value, Start subscript, Start base, x , base End,Start subscript, 0 , subscript End , subscript End , target value End,Start expression, f (x) , expression End , Limit from the Right Endlimxmm-template-arrow-right-8x0+fx) or the left/below (denoted Start Limit from the Left, Start variable, x , variable End,Start target value, Start subscript, Start base, x , base End,Start subscript, 0 , subscript End , subscript End , target value End,Start expression, f (x) , expression End , Limit from the Left Endlimxmm-template-arrow-right-8x0-fx). In principle, these can result in different values, and a limit is said to exist if and only if the limits from both above and below are equal: Start Limit, Start variable, x , variable End,Start target value, Start subscript, Start base, x , base End,Start subscript, 0 , subscript End , subscript End , target value End,Start expression, f (x) , expression End , Limit End = Start Limit from the Right, Start variable, x , variable End,Start target value, Start subscript, Start base, x , base End,Start subscript, 0 , subscript End , subscript End , target value End,Start expression, f (x) , expression End , Limit from the Right End = Start Limit from the Left, Start variable, x , variable End,Start target value, Start subscript, Start base, x , base End,Start subscript, 0 , subscript End , subscript End , target value End,Start expression, f (x) , expression End , Limit from the Left Endlimxmm-template-arrow-right-8x0fx=limxmm-template-arrow-right-8x0+fx=limxmm-template-arrow-right-8x0-fx. For multivariate or complex-valued functions, an infinite number of ways to approach a limit point exist, and so these functions must pass more stringent criteria in order for a unique limit value to exist.

In addition to the formal definition, there are other methods that aid in the computation of limits. For example, algebraic simplification can be used to eliminate rational singularities that appear in both the numerator and denominator, and l'Hôpital's rule is used when encountering indeterminate limits, which appear in the form of an irreducible Start Fraction, Start numerator, 0 , numerator End,Start denominator, 0 , denominator End , Fraction End00  or Start Fraction, Start numerator, ∞ , numerator End,Start denominator, ∞ , denominator End , Fraction End∞∞ .

How Wolfram|Alpha solves limit problems

Wolfram|Alpha calls Mathematica's built-in function Limit to perform the computation, which doesn't necessarily perform the computation the same as a human would. Usually, the Limit function uses powerful, general algorithms that often involve very sophisticated math.

In addition to this, understanding how a human would take limits and reproducing human-readable steps is critical, and thanks to our step-by-step functionality, Wolfram|Alpha can also demonstrate the techniques that a person would use to compute limits. Wolfram|Alpha employs such methods as l'Hôpital's rule, the squeeze theorem, the composition of limits and the algebra of limits to show in an understandable manner how to compute limits.