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
$$f\text{'}(x)=\underset{h\to 0}{\mathrm{lim}}\frac{f(x+h)f(x)}{h},$$
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.
You can also calculate onesided 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. Prior to R2016a, use ezplot
instead
of fplot
.
syms x fplot(x/abs(x), [1 1], 'ShowPoles', 'off')
To calculate the limit as x approaches 0 from the left,
$$\underset{x\to {0}^{}}{\mathrm{lim}}\frac{x}{\leftx\right},$$
enter
syms x limit(x/abs(x), x, 0, 'left')
ans = 1
To calculate the limit as x approaches 0 from the right,
$$\underset{x\to {0}^{+}}{\mathrm{lim}}\frac{x}{\leftx\right}=1,$$
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 

$$\underset{x\to 0}{\mathrm{lim}}f(x)$$ 

$$\underset{x\to a}{\mathrm{lim}}f(x)$$ 

$$\underset{x\to a}{\mathrm{lim}}f(x)$$ 

$$\underset{x\to a+}{\mathrm{lim}}f(x)$$ 
