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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 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 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.

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  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.
  
  

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  Figure 20: plot of sin(x) / x

  
  

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    >> 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.
  
  
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  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.
  
  
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  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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