- #1

Felafel

- 171

- 0

## Homework Statement

Find the limit of

##1): \displaystyle \lim_{n \to +\infty}(\frac{f(a+\frac{1}{n})}{f(a)})^{\frac{1}{n}}##

##2) \displaystyle \lim_{x \to a} (\frac{f(x)}{f(a)})^{\frac{1}{ln(x)-ln(a)}}(=1^{\infty})##

I am not quite sure if i can solve it the way I did, it has been to easy so there's a trick, I'm afraid

## The Attempt at a Solution

##1) \displaystyle \lim_{n \to +\infty}(\frac{f(a+\frac{1}{n})}{f(a)})^{\frac{1}{n}}##

=##\displaystyle \lim_{n \to +\infty} e^{\frac{1}{n} (log(f(a+\frac{1}{n})-log(f(a)))}##=

=##e^0=1##

##2) \displaystyle \lim_{x \to a} (\frac{f(x)}{f(a)})^{\frac{1}{ln(x)-ln(a)}}(=1^{\infty})##

##\displaystyle \lim_{x \to a} e^{ln(\frac{f(x)}{f(a)}) \frac{1}{ln(x)- ln(a)}}##

##\displaystyle \lim_{x \to a} \frac{ln(f(x))-ln(f(a))}{ln(x)- ln(a)}(=\frac{0}{0})## with de l'Hopital

##\displaystyle \lim_{x \to a} \frac{f'(x)}{xf(x)}=\frac{f'(a)}{af(a)}##