# limit

Limit of symbolic expression

## Description

example

limit(f,var,a) returns the Bidirectional Limit of the symbolic expression f when var approaches a.

limit(f,a) uses the default variable found by symvar.

limit(f) returns the limit at 0.

example

limit(f,var,a,'left') returns the Left Side Limit of f as var approaches a.

example

limit(f,var,a,'right') returns the Right Side Limit of f as var approaches a.

## Examples

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Calculate the bidirectional limit of this symbolic expression as x approaches 0.

syms x h
f = sin(x)/x;
limit(f,x,0)
ans = $1$

Calculate the limit of this expression as h approaches 0.

f = (sin(x+h)-sin(x))/h;
limit(f,h,0)
ans = $\mathrm{cos}\left(x\right)$

Calculate the right and left-sided limits of symbolic expressions.

syms x
f = 1/x;
limit(f,x,0,'right')
ans = $\infty$
limit(f,x,0,'left')
ans = $-\infty$

Since the limit from the left does not equal the limit from the right, the two-sided limit does not exist. In this case, limit returns NaN (not a number).

limit(f,x,0)
ans = $\mathrm{NaN}$

Calculate the limit of expressions in a symbolic vector. limit acts element-wise on the vector.

syms x a
V = [(1+a/x)^x exp(-x)];
limit(V,x,Inf)
ans = $\left(\begin{array}{cc}{\mathrm{e}}^{a}& 0\end{array}\right)$

## Input Arguments

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Input, specified as a symbolic expression, function, vector, or matrix.

Independent variable, specified as a symbolic variable. If you do not specify var, then symvar determines the independent variable.

Limit point, specified as a number or a symbolic number, variable, or expression.

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### Bidirectional Limit

$L=\underset{x\to a}{\mathrm{lim}}f\left(x\right),x-a\in ℝ\text{\}\left\{0\right\}.$

### Left Side Limit

$L=\underset{x\to {a}^{-}}{\mathrm{lim}}f\left(x\right),x-a<0.$

### Right Side Limit

$L=\underset{x\to {a}^{+}}{\mathrm{lim}}f\left(x\right),x-a>0.$

## Version History

Introduced before R2006a