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

$f' (x) = \lim_{h \to 0} \frac{f (x + h) - f (x)}{h},$

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')

Figure contains an axes object. The axes object contains an object of type functionline.

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

$\lim_{x \to 0^{-}} \frac{x}{| x |},$

enter

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

ans =
 -1

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

$\lim_{x \to 0^{+}} \frac{x}{| x |} = 1,$

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
$\lim_{x \to 0} f (x)$	`limit(f)`
$\lim_{x \to a} f (x)$	`limit(f, x, a) or` `limit(f, a)`
$\lim_{x \to a -} f (x)$	`limit(f, x, a, 'left')`
$\lim_{x \to a +} f (x)$	`limit(f, x, a, 'right')`