On-line Math 21

1.5  Precise Definition of a Limit

Example 5 Show that

lim x® ¥  2x+3 x-1 = 2.

Hints

Assume that e > 0 is given. We need to be able to get

2x+3 x-1

within e of 2, by taking x large enough. However, that distance can be re-written in such a way that it involves the size of x (this, or analogous tricks, are standard for these calculations).

More

ê ê ê 2x+3 x-1 -2 ê ê ê = ê ê ê (2x+3)-2(x-1) x-1 ê ê ê = ê ê ê 5 x-1 ê ê ê .

Then,

ê ê ê 2x+3 x-1 -2 ê ê ê < e Û   ê ê ê 5 x-1 ê ê ê < e,

which means that

1 |x-1| < e 5 ,

or, solving for x , and taking x > 0 (since x® ¥, this is not a severe restriction):

|x-1| > 5 e  Û  x > 5 e +1

So, take

D = 5 e +1.

Then, if x > D , |f(x)-2| < e, as required.

Copyright (c) 2000 by David L. Johnson.