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Prove the quotient theorem using the limit definition?

  1. Nov 24, 2003 #1
    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).
     
  2. jcsd
  3. Nov 24, 2003 #2
    [tex]
    \begin{equation*}\begin{split}
    \lim_{h\rightarrow 0} \frac {\frac {f(x+h)} {g(x+h)} - \frac {f(x)} {g(x)}} {h}
    &= \lim_{h\rightarrow 0} \frac {f(x+h)g(x) - f(x)g(x+h)} {hg(x+h)g(x)} \\
    &= \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)}
    \end{split}\end{equation*}
    [/tex]
    You can take it from here. Be careful. How do you know
    [tex]\lim_{h\rightarrow0} g(x+h)=g(x)[/tex]?
     
    Last edited: Nov 24, 2003
  4. Nov 25, 2003 #3
    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!
     
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