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 ofcos(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 ofx/|x|, whose graph is shown in the following figure, asxapproaches 0 from the left or from the right.

symsxfplot(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^®returnsNaN(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限制(f,x,0). Explore the options for thelimitcommand in this table, wherefis a function of the symbolic objectx.

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