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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):
           -0.10000000000000   0.99833416646828
           -0.01000000000000   0.99998333341667
           -0.00100000000000   0.99999983333334
           -0.00010000000000   0.99999999833333
           -0.00001000000000   0.99999999998333
           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).

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