Documentation

This is machine translation

Translated by Microsoft
Mouse over text to see original. Click the button below to return to the English verison of the page.

Limits

The fundamental idea in calculus is to make calculations on functions as a variable "gets close to" or approaches a certain value. Recall that the definition of the derivative is given by a limit

f'(x)=limh0f(x+h)f(x)h,

provided this limit exists. Symbolic Math Toolbox™ software enables you to calculate the limits of functions directly. The commands

syms h n x
limit((cos(x+h) - cos(x))/h, h, 0)

which return

ans =
-sin(x)

and

limit((1 + x/n)^n, n, inf)

which returns

ans =
exp(x)

illustrate two of the most important limits in mathematics: the derivative (in this case of cos(x)) and the exponential function.

One-Sided Limits

You can also calculate one-sided limits with Symbolic Math Toolbox software. For example, you can calculate the limit of x/|x|, whose graph is shown in the following figure, as x approaches 0 from the left or from the right.

syms x
fplot(x/abs(x), [-1 1], 'ShowPoles', 'off')

To calculate the limit as x approaches 0 from the left,

limx0x|x|,

enter

syms x
limit(x/abs(x), x, 0, 'left')
ans =
 -1

To calculate the limit as x approaches 0 from the right,

limx0+x|x|=1,

enter

syms x
limit(x/abs(x), x, 0, 'right')
ans =
1

Since the limit from the left does not equal the limit from the right, the two- sided limit does not exist. In the case of undefined limits, MATLAB® returns NaN (not a number). For example,

syms x
limit(x/abs(x), x, 0)

returns

ans =
NaN

Observe that the default case, limit(f) is the same as limit(f,x,0). Explore the options for the limit command in this table, where f is a function of the symbolic object x.

Mathematical Operation

MATLAB Command

limx0f(x)

limit(f)

limxaf(x)

limit(f, x, a) or

limit(f, a)

limxaf(x)

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

limxa+f(x)

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

Was this topic helpful?