limit

Limit of symbolic expression

Syntax

limit(f,var,a)

limit(f,a)

limit(f)

limit(f,var,a,'left')

limit(f,var,a,'right')

Description

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.

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

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

Examples

Limit of Symbolic Expression

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 = $$\cos \left(x\right)$$

Right and Left Limits of Symbolic Expression

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 = $$\mathrm{NaN}$$

Limit of Expressions in Symbolic Vector

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

f — Input

symbolic expression | symbolic function | symbolic vector | symbolic matrix

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

var — Independent variable

x (default) | symbolic variable

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

a — Limit point

number | symbolic number | symbolic variable | symbolic expression

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

More About

Bidirectional Limit

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

Left Side Limit

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

Right Side Limit

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

Version History

Introduced before R2006a