TITLE: $\lim \limits_{h\to \infty} \left(\frac{f(10+\frac{1}{h})}{f(10)}\right)^h $ given that $ f(10) = 5, f'(10)=2$
QUESTION [7 upvotes]: Trying to find this limit: 
$\lim \limits_{h\to \infty} \left(\frac{f(10+\frac{1}{h})}{f(10)}\right)^h $
given that $ f(10) = 5, f'(10)=2$.
Tried: take the log of the limit, then it becomes 
$\lim \limits_{h\to \infty} h\left(\log\left(f(10+\frac{1}{h})\right)-\log f(10)\right) $
I couldn't find a way to make use of  $f'(10)=2$...
Any hint?

REPLY [2 votes]: By Taylor,
$f(10+\frac{1}{h})
=f(10)+\frac{f'(10)}{h}+O(\frac1{h^2})
$
so
$\left(\frac{f(10+\frac{1}{h})}{f(10)}\right)
=\left(\frac{f(10)+\frac{f'(10)}{h}+O(\frac1{h^2})}{f(10)}\right)
=1+\frac{f'(10)}{10h}+O(\frac1{h^2})
$.
Since
$(1+a/h)^h
\to e^a
$
as $h \to \infty$,
$(1+\frac{f'(10)}{10h}+O(\frac1{h^2}))^h
\to e^{f'(10)/10)}
$.