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

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

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')

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

$$\underset{x\to 0^{-}}{\lim }\frac{x}{\left|x\right|},$$

enter

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

ans =
 -1

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

$$\underset{x\to 0^{+}}{\lim }\frac{x}{\left|x\right|}=1,$$

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