On-line Math 21

1.4  Computing limits

Example 2 Compute

lim x® 3  x2-x-6 x3-4x2+3x .

Hints

For this, we note first that when x = 3, both the numerator and the denominator are 0. This is, as mentioned earlier, a signal that you have to do more. What can you do to make sense of that limit? Certainly 0/0 is not a good answer.

Another hint

The key to making sense of the 0/0 in this limit is to factor both the numerator and denominator. That may seem hard, but remember that both the numerator and the denominator are 0 when x = 3 , so (x-3) is a factor of both:

x2-x-6 x3-4x2+3x = (x-3)() (x-3)() .

Still stuck?

x2-x-6 x3-4x2+3x = (x-3)(x+2) (x-3)(x)(x-1) = (x+2) (x)(x-1) ,

so,

lim x® 3  x2-x-6 x3-4x2+3x = lim x® 3  x+2 x(x-1) = 3+2 3(3-1) = 5 6 .

Copyright (c) 2000 by David L. Johnson.