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If $\lim_{x\to a}f(x)=1$ and $\lim_{x\to a}g(x)=\infty$ then show that $$\lim_{x\to a}\{f(x)\}^{g(x)}=e^{\lim\limits_{x\to a}{g(x)\{f(x)-1\}}}.$$ I started off as:

$$\lim_{x\to a}\{f(x)\}^{g(x)} = e^{\lim\limits_{x\to a}{g(x)\{\ln f(x)\}}}$$ $$=e^{\lim\limits_{x\to a}{\{\ln f(x)\}\over {\frac{1}{g(x)}}}}$$ $$=e^{-\lim\limits_{x\to a}{\frac{f'(x)}{g'(x)}.\frac{\{g(x)\}^2}{f(x)}}}.$$

But this doesn't seem to be going anywhere.

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3 Answers 3

up vote 2 down vote accepted

$$\begin{align} \lim_{x\rightarrow\infty}f(x)^{g(x)}&=\lim_{x\rightarrow\infty}\left[1+\frac1{\frac1{f(x)-1}}\right]^{g(x)}\\ &=\lim_{x\rightarrow\infty}\left[1+\frac1{\frac1{f(x)-1}}\right]^{\frac1{f(x)-1} \cdot(f(x)-1)g(x)}\\ &=\lim_{x\rightarrow\infty} e^{(f(x)-1)g(x)}\phantom{\frac1{\frac1n}} \end{align}$$

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Latex issue: Does anyone know how to make the lines to be more separate? vspace doesn't seem to work. –  ajotatxe Apr 26 at 9:34
    
\phantom{blah} produces a blank area of the same size as its argument would take up if typeset. Normally, that's used for horizontal alignment but I used it in your example to give a bit more vertical space. –  David Richerby Apr 26 at 12:41
    
Actually, \vphantom would have been a better way but, hey, \phantom worked. –  David Richerby Apr 29 at 8:23
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Hint: use $\lim\limits_{x \to 0}\frac{\ln (1+x)}{x}=1$.

Then $\lim\limits_{x \to a} \ln f(x) = \lim\limits_{x \to a} \left( \frac{\ln (1+[f(x)-1])}{f(x)-1}\cdot[f(x)-1] \right)=...?$

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If

$\lim_{x→α}f(x)=1$ and $\lim_{x→α}g(x)= \infty $

then $$\lim_{x→α}(f(x))^{g(x)}= \lim_{x→α}(1+f(x)−1)^{g(x)}= \lim_{x→α}[[(1+f(x)−1)^{\frac{1}{f(x)-1}}]^{(f(x)-1)}]^{g(x)}= \lim_{x→α}[(1+f(x)−1)^{\frac{1}{f(x)-1}}]^{\lim_{x→α}(f(x)-1)g(x)}= e^{\lim_{x→α}(f(x)-1)g(x)}.$$

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See also math.stackexchange.com/questions/632527/… –  medicu Apr 26 at 14:36
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