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.
That's pretty straight forward. In terms of the definitions of those limits you want to prove that
"Given there exist such that if then "
if and only if
"Given there exist such that if then .
(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 . In other words, let .