0
$\begingroup$

Under which conditions

$$\lim_{a\to+\infty}\ln(f(a,x)) = \ln(z(x))\Longrightarrow \underset{a\to+\infty}{\lim}f(a,x) = z(x)\;?$$

  • 1
    Was the switching of the parameters $a,x$ in $f$ in the second limit intentional?2012-10-21
  • 0
    No, it should read as $f(a,x)$ or $f(x,a)$ in both cases.2012-10-21
  • 0
    Put it simpler I just want to know if I find a limit of log of something is equal to the log. Can I infer that limit of the expressions under logs are equal.2012-10-21
  • 0
    Or it would be simpler if the following is true $\lim_{a\to+\infty}\ln(f(a,x)) = \ln(\lim_{a\to+\infty} f(a,x))$. Is it always true and why?2012-10-21

1 Answers 1

1

It's always true. You can take exponentials on both sides, and then pull the $\lim$ outside on the left-hand side since the exponential function is continuous.

  • 0
    So can we claim that $$\lim_{a\to+\infty}g(f(a,x)) = g(z(x))\Longrightarrow \underset{a\to+\infty}{\lim}f(a,x) = z(x)\;?$$ for any continuous function $g(.)$? Which book should I consult to understand why this is so?2012-10-21
  • 0
    @arkadiy: No, certainly not! But it's true for any invertible function with continuous inverse, since then you can apply $g^{-1}$ to both sides, etc. You don't really need to consult a book, since that's all there is to this argument, but any calculus textbook should do if you insist. ;-)2012-10-21
  • 0
    Now I got it. Thank you very much! No need for a calculus textbook.;)2012-10-21