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