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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.

next up previous
Next: Limits of more Up: Background Previous: Background

William W. Farr
Thu Sep 7 15:27:05 EDT 1995