second to last question.

Prove that lim_{x to x_0} f(x) = L if and only if lim_{x to 0} f(x+x_0) = L. Assume that L is finite.

- Feb 28th 2010, 06:42 PMtn11631Prove that lim_{x to x_0} f(x) = L if and only if lim_{x to 0} f(x+x_0) = L. Assume
second to last question.

Prove that lim_{x to x_0} f(x) = L if and only if lim_{x to 0} f(x+x_0) = L. Assume that L is finite. - Mar 1st 2010, 03:17 AMHallsofIvy
That's pretty straight forward. In terms of the

**definitions**of those limits you want to prove that

"Given $\displaystyle \epsilon> 0$ there exist $\displaystyle \delta> 0$ such that if $\displaystyle |x-x_0|< \delta$ then $\displaystyle |f(x)- L|< \epsilon$"

if and only if

"Given $\displaystyle \epsilon> 0$ there exist $\displaystyle \delta> 0$ such that if $\displaystyle |y|< \delta$ then $\displaystyle |f(x_0+ y)- L|< \epsilon$.

(I've changed "x" to "y" in the second part so as not to confuse the two uses of "x".)

Comparing those two, they will be the same if $\displaystyle x_0+ y= x$. In other words, let $\displaystyle y= x- x_0$. - Mar 1st 2010, 08:53 AMtn11631
In other words, let y= x- x_0.....do i have to show anything more or do the two definitions take of it? Sorry i'm really bad at proofs..

- Mar 1st 2010, 12:46 PMDrexel28