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

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
More examples
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

Specify the direction of approach:

limit tan(t) as t->pi/2 from the left
More examples
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
More examples
Limit Representations

Express functions in terms of limits.

Find limit representations for a function:

exp(z) limit representation
Si(x) limit rep
More examples
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)
More examples
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}
More examples