Existence of Limit

• Dec 25th 2005, 01:05 PM
ThePerfectHacker
Existence of Limit
Let $\displaystyle 0 \leq f(x)$
Then proof existence of infinity or a limit (but not both):
Thus,
$\displaystyle \lim c \rightarrow \infty f(x)=L$
or
$\displaystyle \lim c \rightarrow \infty f(x)=\infty$

Logic caution:
In the problem you have to proof that exactly one of the conditions must be satisfied and exactly one.
• Dec 25th 2005, 01:20 PM
CaptainBlack
Quote:

Originally Posted by ThePerfectHacker
Let $\displaystyle 0 \leq f(x)$
Then proof existence of infinity or a limit (but not both):
Thus,
$\displaystyle \lim c \rightarrow \infty f(x)=L$
or
$\displaystyle \lim c \rightarrow \infty f(x)=\infty$

Logic caution:
In the problem you have to proof that exactly one of the conditions must be satisfied and exactly one.

Let $\displaystyle f(x)=\sin(x)+1$, then what is (if anything):

$\displaystyle \lim_{x \rightarrow \infty}f(x)\ ?$

Or have I misunderstood your intention?
• Dec 25th 2005, 03:59 PM
ThePerfectHacker
Yes, I made a mistake with what I said you are correct.
Show that if the limit is L then it cannot be infinite.
Show that if the limit is infinite then it cannot be L.
• Dec 26th 2005, 12:46 PM
mooshazz
Quote:

Originally Posted by ThePerfectHacker
Yes, I made a mistake with what I said you are correct.
Show that if the limit is L then it cannot be infinite.
Show that if the limit is infinite then it cannot be L.

in case of infinity then for any given M there is X that for every x>X, f(x)>M
take M to be L+100 and L is not a limit

in case of L for every given g>0 there is X wich for every x>X, |f(x)-L|<g
take X to be the one that from him forward f(x) is blocked (don't remember the english expression) (and don't remember the proof that you have one) in this part f(x)<>infinity