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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):
    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).

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