Limits

## 8   Limits

Concept: Definition of a limit.
The notion of a limit of a function was originally introduced in order to make the definition of derivative make sense. However, it can also be used to describe the behavior of a function near a given point, as well as to find and describe asymptote s.

One needs to compute limits of the form limx ® a f(x) to compute the slope of a tangent line or the derivative of a function, as well as to determine whether a function is continuous.

The definition of a limit of a function, which can be found in any calculus textbook, forms the basis for both a graphical and numerical approach to finding limits.

To summarize it in a means suitable for MATLAB exploration, one has as the value of x gets close to a desired number (often called c), then the value of f(x) gets close to a number L called the limit.

If we want to check out a limit with MATLAB we need to understand this graphically and numerically. Graphically it says if we follow the graph of f(x) towards c from the left or the right, the corresponding y values converge to a limit. Numerically, this says if we look at the sequence of y values corresponding to any sequence of x values which gets close to c then the y values should get very close to a single number -- the limit.

A key result we'll need is that limx ® a f(x) = L if and only if both limx ® a- f(x) = L and limx ® a+ f(x) = L. In other words, we have the following.

The two-sided limit exists if and only if the left- and right-sided limits both exist and are equal.

Thus, to show that the limit is L, essentially we need to show that, as x takes values closer to a (on both sides of a), f(x) takes values closer to L.

• Example: Finding Limits Graphically

Let us use a graphical approach to determine limx ® 0 sin x/x.

Notice this function is not defined at x=0, and plugging in'' x=0 gives an indeterminate form of 0/0. Thus the limit takes work to figure out. However, this function is defined for all x except at x=0, which is all that is required to apply the limit definition. The following MATLAB commands will plot the graph of sin x / x near x=0.

Figure 20: plot of sin(x) / x

  >> x=linspace(-1,1);    % plot for x in the interval [-1, 1]
>> y=sin(x)./x;         % dot' makes division point wise
>> plot(x,y), grid

From the graph you notice as one moves toward 0 from either side on the x-axis, the y values move toward 1. From the graph you can tell what the limit is 1.

• Example: Finding Limits Numerically

Here is a simple method for finding the same limit numerically.

>> format long                         % gives extra decimal places
>> x=[-.1 -.01 -.001 -.0001 -.00001]   % x approaches 0 from the left
>> y=sin(x)./x; [x;y]'                 % print x,y in a 2-column table
>> x=[.1 .01 .001 .0001 .00001]        % x approaches 0 from the right
>> y=sin(x)./x; [x:y]'

You should get the following results (x on the left, sin (x)/x on the right):
ans=
-0.10000000000000   0.99833416646828
-0.01000000000000   0.99998333341667
-0.00100000000000   0.99999983333334
-0.00010000000000   0.99999999833333
-0.00001000000000   0.99999999998333

ans=
0.10000000000000   0.99833416646828
0.01000000000000   0.99998333341667
0.00100000000000   0.99999983333334
0.00010000000000   0.99999999833333
0.00001000000000   0.99999999998333


Since the values in the right columns both approach 1, we conclude that both right and left limits are 1 and so the limit is 1 as well.

• Example: limit of slope of the secant line:
See the secant line example to find the definition of the secant line. Essentially we have a function f(x) a point x1 and a point x2. Let x2 = x1 + h. So that h measures the difference between the two values of x. Then as h gets close to 0 x2 and x1 get close to each other. If the limit as h goes to zero exists then the function is said to have a derivative at x1.

In the secant line example we see how to plot the function sin(x) and the secant line between x1 = p/4 and x2 = x1 + h with h = p/8. By changing the value of h we see it is a simple matter to find graphically the limit of the secant lines.

To find the slope numerically we recall the definition of the slope of a line
m = (f(x1+h) - f(x1))/(x1 + h - x1) = (f(x1+h) - f(x1))/ h
So in MATLAB we define a sequence of values h converging to 0, and then plug into the formula:
>> x_1=pi/4; n = 1:5; h = 10.^(-n);     % suppress the output with ;
>> m = ( sin(x_1 + h) - sin(x_1)) ./ h; % we need the ./ as h is a
% list
>> [h;m]'                               %  print out in columns
`
If you repeat this using -h in the second and 3rd line you will see that the left and right limits are the same and equal 0.70710 (=2/2).