On-line Math 21

1.5  Precise Definition of a Limit

Example 3 Show that

lim x® 3  1 x = 1 3 .

Hints

Again, you need to do a ballpark estimate. But, first you want to look at what you are trying to control, and re-express it to make |x-3| appear.

Next step

ê ê ê 1 x - 1 3 ê ê ê = |3-x| |3x| .

And then?

ê ê ê 1 x - 1 3 ê ê ê = |3-x| |3x| = 1 3|x| |x-3|.

We need to control the 1/3|x| term, since the other one we have complete control over.

Finish it.

Copyright (c) 2000 by David L. Johnson.