# function proof

• Oct 20th 2007, 02:18 PM
ruprotein
function proof
(2) Show that lim_{x \to x_0} f(x) = L if and only if lim_{x \to 0} f(x+x_0) = L. Assume x_0 and L are finite.
(3) Show that if lim_{x \to x_0} f(x) = L and E is a set which has x_0 as an accumulation point, then lim_{x \to x_0, x in E} f(x) = L. Give an example to show that the converse may fail. Assume x_0 and L are finite.

can anyone solve any of these?
• Oct 20th 2007, 03:47 PM
Plato
You know that if you could post using even basic LaTeX, we could understand what you are asking. It is very hard to know what you are asking.
• Oct 20th 2007, 03:58 PM
ruprotein
what do u mean basic latex?
• Oct 20th 2007, 04:04 PM
Jhevon
Here is what you asked in basic LaTex. Incidentally, what you typed was very close to LaTex code, add a few "\"'s and put math tags around it and you would have:

Quote:

Originally Posted by ruprotein
(2) Show that $\displaystyle \lim_{x \to x_0} f(x) = L$ if and only if $\displaystyle \lim_{x \to 0} f(x+x_0) = L$. Assume $\displaystyle x_0$ and $\displaystyle L$ are finite.

(3) Show that if $\displaystyle \lim_{x \to x_0} f(x) = L$ and $\displaystyle E$ is a set which has $\displaystyle x_0$ as an accumulation point, then $\displaystyle \lim_{x \to x_0} f(x) = L$, $\displaystyle x \in E$. Give an example to show that the converse may fail. Assume $\displaystyle x_0$ and $\displaystyle L$ are finite.

can anyone solve any of these?

• Oct 20th 2007, 04:58 PM
ThePerfectHacker
Quote:

Originally Posted by ruprotein
(2) Show that lim_{x \to x_0} f(x) = L if and only if lim_{x \to 0} f(x+x_0) = L. Assume x_0 and L are finite.

$\displaystyle \lim_{x\to x_0} f(x) = L$ means:
$\displaystyle 0<|y-x_0| < \delta \implies |f(x) - L| < \epsilon$.
Now if $\displaystyle y = x_0+x$ where $\displaystyle 0<|x|<\delta$ we have:
$\displaystyle |y-x_0| = |x_0+x - x_0| = |x| \in \left( 0, \delta \right)$
Thus,
$\displaystyle |f(x+x_0) - L| < \epsilon$.
• Oct 21st 2007, 08:44 PM
ruprotein
thanks a lot has anyone ghot the other one?

Show if lim f(x) = L then if E is any set which has x_o as an accumulation point , lim f(x) = L. But show converse fails.
• Oct 22nd 2007, 09:55 AM
ThePerfectHacker
Quote:

Originally Posted by ruprotein
thanks a lot has anyone ghot the other one?

Show if lim f(x) = L then if E is any set which has x_o as an accumulation point , lim f(x) = L. But show converse fails.

Is $\displaystyle E = \{ f(x) \}$?
• Oct 22nd 2007, 11:31 AM
ruprotein
the guy above restated the question in Latex form