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
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)
ans = -sin(x)
limit((1 + x/n)^n, n, inf)
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 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 ezplot(x/abs(x), -1, 1)
To calculate the limit as x approaches 0 from the left,
syms x limit(x/abs(x), x, 0, 'left')
ans = -1
To calculate the limit as x approaches 0 from the right,
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)
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.