In this section, I'll discuss proofs for limits of the form . They are like proofs, though the setup and algebra are a little different.

Recall that means that for every , there is a such that if

* Definition.* means that for every , there is an M such that if

In other words, I can make as close to L as I please by making x sufficiently large.

* Remarks.* Limits at infinity often occur as * limits of sequences*, such as

In this case, . I won't make a distinction between the limit at infinity of a sequence and the limit at infinity of a function; the proofs you do are essentially the same in both cases.

There is s similar definition for , and the proofs are similar as well. I'll stick to here.

* Example.* Prove that .

As with proofs, I do some scratch work, working backwards from what I want. Then I write the "real proof" in the forward direction.

* Scratch work.* I want

I want to drop the absolute values, so I'll assume . Rearranging the inequality, I get .

Here's the real proof. Let . Set . Since , I have . Suppose . Then , and

This proves that .

* Example.* Prove that .

* Scratch work.* I want

In order to drop the absolute values, I need to assume .

Rearrange the inequality:

Here's the real proof. Let . Set . If , then and . So

Therefore,

Note that the expression would be negative if . So I took M to be the max of 0 and to ensure that if , then x would be positive. Now you actually need
to be positive in order to put on the
absolute values, and if . It isn't hard to prove that , so in fact I
don't need to take the max with 0 --- provided that I'm willing to
*prove* that . I decided to take the easy way out!

* Example.* Prove that is undefined.

I'll use proof by contradiction. Suppose that

Taking in the definition, I can find M such that if , then .

Choose p to be an even number greater than M. Then

This says that the distance from L to 1 is less than , so

Choose q to be an odd number greater than M. Then

This says that the distance from L to -1 is less than , so

This is a contradiction, since L can't be in and in at the same time.

Hence, is undefined.

Copyright 2018 by Bruce Ikenaga