On-line Math 21

Example 5 Let

f(x) = x3+1 x2-1 .

Then

lim x® -1  f(x) = -3 2 .

Solution

This one is more subtle. The problem with this limit, as with the one discussed at first, is that both the numerator and the denominator go to 0 as x goes to -1 . This is what is called an indeterminate form, and in general is a signal that you have more work to do to figure the limit out. In this case we can see why the limit is -3/2 by simplifying the expression for f(x) :

f(x) = x3+1 x2-1 = (x2-x+1)(x+1) (x-1)(x+1) = (x2-x+1) (x-1) ,

so

lim x® -1  f(x) = lim x® -1  (x2-x+1) (x-1) = ((-1)2-(-1)+1) ((-1)-1) = 3 -2 = - 3 2 .

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Copyright (c) 2000 by David L. Johnson.