The limit of a function is a fundamental concept in calculus and analysis concerning the behaviour of that function near a particular input.

In this activity we will use the commands syms and limit to calculate limits of real functions. Furthermore, we will learn how to plot piecewise functions using the commands piecewise and fplot.

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MATLAB provides the limit command for calculating limits. In its most basic form, the limit command takes an expression as an argument and finds the limit of this expression as the independent variable goes to zero.

For example, let us calculate the limit

The limit command falls in the realm of symbolic computing; you need to use the syms command to tell MATLAB which symbolic variables you are using. That is:

syms x % Define symbolic variable

limit((x^3 + 5)/(x^4 + 7)) % Calculate limit

You can also compute limit of a function, as the variable tends to some number other than zero. In general, to calculate

we use the command limit(f, x, a) where "f" is the expression, "x" is the variable and "a" is the number that "x" is approaching.

For example, let us calculate:

- ; and
- .

Thus, we write:

syms x % Define symbolic variable "x"

limit(x^2 + 5, x, 3)

limit((2 * x - 3)/(x - 1), x, 1)

Remark: If the f depends on only one variable, we can omit it in the code, e.g. limit(x^2 + 5, 3).

Also notice that the output of limit((2 * x - 3/(x - 1)) is NaN (not a number). This means that the limit does not exist! We can appreciate this by analysing the plot the graph of using the command fplot as follows:

syms x % Define symbolic variable "x"

g0 = (2 * x - 3)/(x - 1); % Define a variable "g0" in terms "x"

fplot(x, g0) % Plot the function

xlabel('x') % Now we just add some settings to the plot

ylabel('g(x)=(2x-3)/(x-1)')

grid on

axis([-4 4 -8 8])

Run this section and analyse the graph of . What do you notice?

Another way to confirm that the limit does not exist is by calculating one-sided limits.

We can also calculate the left-handed and right-handed limits. These are computed in MATLAB by passing the character strings 'left' and 'right' to the limit command as the last argument. For example:

syms x

g0 = (2 * x - 3)/(x - 1)

limit(g0, x, 1,'left')

limit(g0, x, 1,'right')

Remark: In this case we cannot omit the variable "x" in the command limit. Try it yourself! Delete "x" in any of the lines with the limit commad, from the previous code, and re-run this section. You will get an error because MATLAB always requires the symbolic variable to calculate one-sided limits.

In MATLAB we can also calculate limits of piecewise functions. For example, consider the function

Before calculating limits, first let us plot this function on the interval using the commands piecewise and fplot as follows:

syms x

f0 = piecewise(x < 0, 3/2 * x - 1, 0 <= x, exp(x^2)/(1 + x^2))

fplot(f0,'r')

grid on, xlabel('x'), ylabel('y'), axis([-3,3,-4,4])

Remark: Notice that all the commands in the last line of the previous code are separated with a comma ",". Compare with the code for plotting the example from section 1.2.

Now we can calculate the one-sided limits of . To do this we can make use of the value of the variable f0, defined above. Thus, we have:

syms x

limit(f0, x, 0, 'left')

limit(f0, x, 0, 'right')

Questions: Do these values agree with the graph previously plotted? Is this function continuous or discontinuous at ?

MATLAB allows you also to calculate limits approaching to infinity. For example, in order to calculate

we just need to write:

syms x

func = (2 * x - 3)/(x - 1)

limit(func, x, Inf)

limit(func, x, -Inf)

We can also verify the basic properties of limits using the command limit. For example, let us calculate the limit of the functions

as x tends to 5. To do this we write:

syms x

f = (3 * x - 5)/(x - 3);

g = x^2 + 1;

l1 = limit(f, 4)

l2 = limit (g, 4)

lAdd = limit(f + g, 4)

lSub = limit(f - g, 4)

lMult = limit(f * g, 4)

lDiv = limit (f / g, 4)

Let's practice what we just learned.

Calculate the following limits:

Write your code here:

Use the commands piecewise and fplot to plot the graphs of the following functions on the indicated axes.

a) , with the x-axis raging from to 4 and the y-axis ranging from to 8.

b) , with the x-axis raging from to and the y-axis ranging from to .

Suggestion: Remember to use different variables in your code. You can add labels to identify each graph and change the axis scales.

Write your code for here:

Write your code for here:

Determine if is continuous or discontinuous at . Write your code here:

Determine if is continuous or discontinuous at . Write your code here: