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-side limits of symbolic expressions.

syms x
f = 1/x;
limit(f,x,0,"right")

ans = $$\infty$$

limit(f,x,0,"left")

ans = $$-\infty$$

Because the limit from the left is not equal to 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 = $$\left(\begin{array}{cc}{\mathrm{e}}^a& 0\end{array}\right)$$

Set Assumption When Calculating Limit

Calculate the right-side limit of ${\mathit{x}}^{\mathit{n}}$ as $\mathit{x}\to 0^{+}$ for real $$n$$ and then for positive $$n$$.

Create the symbolic variables x and n. When you create symbolic variables, they are assumed to be complex by default. Set the assumption that n is real.

syms x n
assume(n,"real")

Find the limit using this assumption.

limit(x^n,x,0,"right")

ans = 
$$\{\begin{array}{cl}1& \text{ if }n=0\\ 0& \text{ if }0<n\\ \infty & \text{ if }n<0\end{array}$$

Next, assume that n is positive. Find the limit using this assumption.

assume(n>0)
limit(x^n,x,0,"right")

ans = $$0$$

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 | 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