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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

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

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,

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

enter

syms x
limit(x/abs(x), x, 0, 'left')
ans =
-1

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

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

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

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

limit(f)

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

limit(f, x, a) or

limit(f, a)

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

limit(f, x, a, 'left')

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

limit(f, x, a, 'right')

Mathematical Modeling with Symbolic Math Toolbox

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