Prove the quotient theorem using the limit definition?

[franz32]

Hello guys!

I'm new here! Well, it feels like this forum is cool and interesting!

Can anyone help me here? =)

How do you prove the quotient theorem using the limit definition?
(Given a limit of f of x as x approaches a is A and a limit of g of x as x approaces a is B).

 

[StephenPrivitera]

$$\begin{equation*}\begin{split}<br /> \lim_{h\rightarrow 0} \frac {\frac {f(x+h)} {g(x+h)} - \frac {f(x)} {g(x)}} {h} <br /> &= \lim_{h\rightarrow 0} \frac {f(x+h)g(x) - f(x)g(x+h)} {hg(x+h)g(x)} \\<br /> &= \lim_{h\rightarrow 0} \frac {f(x+h)g(x) - f(x)g(x) + f(x)g(x) - f(x)g(x+h)} {hg(x+h)g(x)}<br /> \end{split}\end{equation*}$$
You can take it from here. Be careful. How do you know
$$\lim_{h\rightarrow0} g(x+h)=g(x)$$?

 

[franz32]

Thank you... =)

Hello. =)

Well, I think I can take it from here. If I have doubts,

I would probably want to clarify it.

Anyway, thank you very much!