On-line Math 21

On-line Math 21

1.3  Infinite Limits

Infinity ( ¥) occurs many times in calculus. It's not just some sort of science-fiction invention, it has meaning, in terms of limits. In the case of

lim
x® ¥ 
 f(x) = L

it means that, as x gets larger without bound, f(x) settles down and gets closer to L .

*3in1.5inFigure

Similarly,

lim
x® ¥ 
 f(x) = ¥

means that, as x gets larger and larger, so does f(x) .

Finally, it might happen that a function might itself get large without bound, as x goes to a real number a . We then say that

lim
x® a 
 f(x) = ¥,
or that f has a vertical asymptote at x = a , which might look like one of these graphs

As you might guess, the stuff about one-sided limits [Link] applies here as well. Also, you can be more specific, in some cases, and talk about limts as x goes to +¥ or -¥, or limits being +¥ or -¥. This has specifirc implications on the general ``shape'' of the function, and can help draw the graph y = f(x) .

None of this explanation really offers much insight on how you can compute limits of more than the simplist examples. There are a few standard theorems and techniques that make these computations straightforward.

Example 1

lim
x® ¥ 

Ö
 
x2+1
 
 -1
x

Answer

Solution

Example 2

lim
x® ¥ 
 2x3+x2+x-6
x3+4x+5

Hints

Example 3

lim
x® 1- 
 1
x-1

Hints

Example 4

lim
x® ¥ 
 x2+5x-6
x3+x+5

Hints

Exercise 2

lim
x® ¥ 
Ö
 
x2+2x-1
 
 -x =

Hint

Exercise 3

lim
x® ¥ 
 x2+5x-6
x3+5
 =

Exercise 4

lim
x® 1+ 
 x2-5x+6
x-1
 =

Exercise 5

lim
x® ¥ 
 x3-3x+6
x2-x+5
 =

Copyright (c) 2000 by David L. Johnson.

File translated from TEX by TTH, version 2.61.
On 12 Oct 2000, 22:55.