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 lim
_{x ® 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 lim
_{x ® a} f(
x) =
L if and only
if
both lim
_{x ® a}^{} f(
x) =
L and lim
_{x
® a}^{+} f(
x) =
L. In other words, we have the
following.
The twosided limit exists if and only if the left and
rightsided 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
lim_{x ® 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 xaxis, 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 2column 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 x_{1} and a point x_{2}. Let x_{2} = x_{1} +
h. So that h measures the difference between the two values of
x. Then as h gets close to 0 x_{2} and x_{1} get close to each
other. If the limit as h goes to zero exists then the function
is said to have a
derivative
at x_{1}.
In the secant line example
we see how to plot the function sin(x) and the secant line
between x_{1} = p/4 and x_{2} = x_{1} + 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(x_{1}+h)  f(x_{1}))/(x_{1} + h  x_{1})
= (f(x_{1}+h)  f(x_{1}))/ 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).