Question

What conditions need to be met for \[\lim _{x \rightarrow c} (\frac{g(x)}{f(x)})= \frac{\lim_{x \rightarrow c} g(x)}{\lim_{x \rightarrow c} f(x)}\] apart from the self-evident \[\lim_{x \rightarrow c} f(x) \ne 0\]?

Answer 1

Are there any other restrictions?

Answer 2

they both have to be continues at c

Answer 3

example\[\lim_{x \rightarrow -1}\frac{\sqrt{x}}{x}=\frac{\sqrt{-1}}{-1}\] error

Answer 4

oh wait not continuas but have a limit at -1

Answer 5

or c

Answer 6

Well, I was talking more generally about complex analysis, so that limit should be fine surely?

Answer 7

Isn't that example just -i?

Answer 8

OK, I'll take continuity, makes sense really. Thank you

Answer 9

can we use the fact that the limit exist instead of continuity since a limit can exist even if the point is not continuous :shortly the \[\lim_{x \rightarrow c^+}f(x)=\lim_{x \rightarrow c^-}f(x)\]

Answer 10

the lim has to exist for both functions