Need help Proving infinite limit properties.

[leoxy520]

Suppose that limx->c f(x) = infinity and limx->c g(x)=l where l is a real number. Prove the following.
limx->c[f(x)+g(x)]= infinity
limx->c[f(x)g(x)]= infinity if l > 0
limx->c[f(x)g(x)]= -infinity if l < 0

I have the proof for these already, but I couldn't understand them, would someone please explain them.
The thing I don't understand is where does the L come from, but explanation in general would be greatly appreciated

Here is the proof for the second one:

[URL=http://imageshack.us/photo/my-images/190/unledmkd.png/][PLAIN]http://img190.imageshack.us/img190/2665/unledmkd.png[/URL]

Uploaded with ImageShack.us[/PLAIN]

 

[mathman]

There seems to be a change of notation. In the original, g -> c when x -> a. In the equations in question, g -> L when x -> c.

 

[leoxy520]

There seems to be a change of notation. In the original, g -> c when x -> a. In the equations in question, g -> L when x -> c.

sorry, i corrected it, does anyone have the explanation for this proof ?

 

[Tomer]

What do you mean "where does the L come from"? It is the limit of the function g(x).

I remind you, what you need to prove, is that given an M>0, you can find a $\delta>0$, so that for every x that satisfied |x-c|<$\delta$,
|f(x)g(x)| > M hold.

The proof shows the existence of such a $\delta>0$.

In order to do this alone, however, you should start from the end. You should ask yourself, how can I make |f(x)g(x)| > M happen?
You need to remember that f(x) can be as big as you want when you're close enough to c, and that g(x) can be as close to L as you'd like when you're close enough to c.

That's exactly what they used in the proof.