Inequalities concerning limits

Hi you guys,

so I have some difficulty understanding some of the content in my textbook. I recently started learning about limits and now I got to the formal defintion of a limit in my textbook which sounds/looks perfectly reasonable to me:

For a function defined in some open interval containing (but not necessarily at itself), we say

,

if given any number , there is another number , such that guarantees that .

Now this is shown in an example where it is to be proved that .

Analagous to the definition it says there must be a for which guarantees that for any given .

Next thing it says is the following:

,

which makes sense since it's just factoring.

Because our sole concern is what's in close proximity to , it is assumed that be in the interval and from that it follows that .

Now my actual problem:

Why is (and that's what the textbook says) ?

That's probably some simple algebra and I'm overlooking something, but I really appreciate your help and thank you in advance.

Greetings

Re: Inequalities concerning limits

Quote:

Originally Posted by

**Floele1106** Now my actual problem:

Why is (and that's what the textbook says)

?

We multiply both sides of the inequality |x + 2| ≤ 5 by |x - 2|. Since |x - 2| is nonnegative, the direction of the inequality does not change.

Re: Inequalities concerning limits

Well, that definitely makes sense. Thanks a lot.