next up previous
Next: Definition of the Up: Labs and Projects for Previous: Linear and Quadratic

Limits of functions.

Purpose

The purpose of this lab is to use Maple to become more familiar with limits of functions, including one-sided limits.

Background

Simple limits and Maple

Limits of many functions and expressions can be computed in Maple with the limit command. Some examples are given below.

  > limit(x^2+2*x,x=2);

  > limit(sin(x)/x,x=0);

  > f := x -> (x+3)/(x^2+7*x+12) ;

  > limit(f(x),x=-3);

  > limit(f(x),x=-4);

If the limit exists, Maple can usually find it. In cases where the limit doesn't exist, Maple gives the answer infinity for an unbounded limit or gives a range like -1..1 if the limit doesn't exist, but the expression or function is bounded. See the examples below.

  > limit(1/x,x=0);

  > limit(sin(1/x),x=0);

You can also use Maple to compute limits as x goes to as shown below.

  > f(x);

  > limit(f(x),x=infinity);

  > limit(f(x),x= -infinity);

The formal definition for a limit is given below.

This definition may seem complicated, but its graphical interpretation is not so bad. It says that if you plot with the y range set to you can always choose a value of small enough so that when you shrink the x plot range to and plot the function, its graph will not intersect the top or the bottom edges of your plot. For example, suppose , a=2 and . Then any value of smaller than about will work. To see what is going on, look at the plots generated by the following commands.

  > f := x -> x^2;

  > limit(f(x),x=2);

  > plot(-0.2,0.2,f(x)-4,x=2-0.1..2+0.1,y=-0.2..0.2);

  > plot(-0.2,0.2,f(x)-4,x=2-0.048..2+0.048,y=-0.2..0.2);

In the first of the two plot commands, the value of is . This is too large, since the graph intersects the lines and . The value of for in the second plot command, however, is small enough, since the graph of goes off the sides of the plot. Make sure that you understand this example. If you don't understand, ask for help.

Limits of more complicated functions

It should be no secret by now that for most functions defined by a single formula, when exists. For more complicated functions, this may not be true. For dealing with some of these exceptional cases, we need to define right-hand and left-hand limits. Loosely speaking, the right-hand limit of at a is L if approaches L as x approaches a from the right. That is, the values of x satisfy x > a. The left-hand limit is defined in an analogous manner, with the values of x approaching a from the left. Maple can compute these special limits with commands like those shown below. The Maple floor function is actually the greatest integer function.

  > plot(floor(x),x=0..4);

  > limit(floor(x),x=1,right);

  > limit(floor(x),x=1,left);

The floor function is one of Maple's defined functions, so you might expect things to work properly. If you define your own function in a piecewise fashion, however, things don't go so smoothly. See the example below.

  > f := x -> if x < 0 then -x else x^2+1 fi ;

f := proc(x) options operator,arrow; if x < 0 then -x else x^2+1 fi end

  > limit(f(x),x=0);

Error, (in f) cannot evaluate boolean

  > limit(f(x),x=0,left);

Error, (in f) cannot evaluate boolean

  > limit(f(x),x=1);

Error, (in f) cannot evaluate boolean

The problem is that Maple can't evaluate the if statement in the definition of unless x has a numerical value. The solution in a case like this is to plot the graph of the function or compute values of the function near a, as shown in the following examples. The quotes in the plot command in the example tell Maple to wait until it has a numerical value for x before it tries to evaluate .

  > plot('f(x)',x=-0.1..0.1);

  > evalf(f(-0.01));

  > evalf(f(-0.0001));

Using these commands as examples, you should be able to figure out that and that .

Exercises

  1. For the functions and values of a given below, go through the following steps.
    i.
    Find whether the exists or not. If it does, determine the limit.
    ii.
    For the limits that exist, find a value of that works for .

    1. .

  2. Find the right- and left-hand limits of the following function at x=1. Also, plot the function and relate your limits to the graph.

    Does exist? Explain your reasoning.

  3. Suppose that the function is defined by

    where a and b are parameters. Can you find values for a and b that will make continuous at x=0? Justify your answer.



next up previous
Next: Definition of the Up: Labs and Projects for Previous: Linear and Quadratic



William W. Farr
Mon Jun 26 13:37:30 EDT 1995