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limit

Limit of symbolic expression

Description

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

example

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.

example

limit(f,var,a,"right") returns the right-side limit of f as var approaches a.

example

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 = cos(x)

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

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

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 = 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 = (ea0)

Calculate the right-side limit of xn as x0+ 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 = 

{1 if  n=00 if  0<n if  n<0

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

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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.

More About

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Bidirectional Limit

L=limxaf(x),xa\{0}.

Left-Side Limit

L=limxaf(x),xa<0.

Right-Side Limit

L=limxa+f(x),xa>0.

Version History

Introduced before R2006a

See Also

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