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Examples for

Limits

Limits are the underlying tool used in calculus, appearing in the definitions of continuity, derivatives and integrals. Wolfram|Alpha has the power to compute bidirectional limits, one-sided limits, supremum and infimum limits, discrete limits and multivariable limits. More information, such as plots and series expansions, is provided to enhance mathematical intuition about a limit.

Limits

Compute limits numerically and symbolically.

Compute a limit:

lim (sin x - x)/x^3 as x->0

Start Limit, Start variable, n , variable End,Start target value, ∞ , target value End,Start expression, Start Power, Start base, 1+ Start Fraction, Start numerator, 1 , numerator End,Start denominator, n , denominator End , Fraction End , base End,Start exponent, n , exponent End , Power End , expression End , Limit Endlimn∞1+1n n

Take the limit of a difference quotient:

lim ((x+h)^5 - x^5)/h as h->0

Compute a limit involving abstract functions:

lim (f(x+h) - 2f(x) + f(x-h)) / h^2 as h->0

Multivariable Limits

Explore limits of expressions in two or more dimensions.

Compute a multidimensional limit:

lim sin(x^2 y)/(x^2+y^2) as (x,y)->(0,0)

lim(x,y) -> (0,0) (x^2-xy+y^2)/(x^2+y^2)

limit x/(x+y^5) as (x,y)->(0,0)

Find the value of a nested limit:

limit arctan(x^2/y^2) as x -> 0 as y -> 0

Discrete Limits

Evaluate limiting behavior of sequences.

Compute discrete limits:

discrete limit of (3 n)/(1 + 6 n) as n -> infinity

discrete lim n/(n + 1)

Compute discrete supremum or infimum limits:

discrete lim inf of (-1)^n as n -> infinity

discrete limsup of sin(pi/3 n) * cos(pi/4 m) as {m, n} -> {infinity, infinity}

One-Sided Limits

Compute one-sided limits from a specified direction.

Compute one-sided limits at a point of discontinuity:

lim (x^2 + 2x + 3)/(x^2 - 2x - 3) as x->3

Start Limit, Start variable, x , variable End,Start target value, 2 , target value End,Start expression, Start Piecewise, Start first value, Start Power, Start base, x , base End,Start exponent, 2 , exponent End , Power End , first value End,Start first condition, x≤2 , first condition End,Start second value, 5x-7 , second value End,Start second condition, x > 2 , second condition End , Piecewise End , expression End , Limit Endlimx2 x2x≤25x-7x > 2

Specify the direction of approach:

Start Limit from the Right, Start variable, x , variable End,Start target value, 0 , target value End,Start expression, Start Fraction, Start numerator, x , numerator End,Start denominator, Start Absolute, Start argument, x , argument End , Absolute End , denominator End , Fraction End , expression End , Limit from the Right Endlimx0+xx

limit tan(t) as t->pi/2 from the left

Limit Representations

Express functions in terms of limits.

Find limit representations for a function:

exp(z) limit representation

Si(x) limit rep

GO FURTHER

Step-by-Step Solutions for Calculus

Calculus Web App

Free Unlimited Calculus Practice Problems

RELATED EXAMPLES

Series Expansions

RELATED WOLFRAM RESOURCES

Wolfram Community Calculus Group

Wolfram Language: Calculus

Supremum & Infimum Limits

Find the limiting upper and lower bounds of functions.

Compute supremum limits:

limsup 1/x, x -> 0

superior limit cos(1/y) as y -> 0

Compute infimum limits:

min lim of sin(x) as x -> infinity

liminf cos(1/x) + sin(x^2)