Question

prove \[\lim_{t\to0}\dfrac{f(x)^t-1}{t}=\ln(f(x))\]

Answer 1

@eliassaab

Answer 2

Looks suspiciously like: \[\Large \frac{y^t-y^0}{t-0}\]

Answer 3

Ahh limit definition of derivative ? nice nice let me try

Answer 4

\[[f(x)^t ]'(0) = f(x)^t \ln f(x) (0) = \ln f(x)\]

\(\blacksquare\) ?

Answer 5

\[\frac{\partial}{\partial y}(f(x)^y)=\lim_{t \to 0} \frac{f(x)^{y+t}-f(x)^y}{t} \\f(x)^y \lim_{t \to 0} \frac{f(x)^t-1}{t} = f(x)^y \ln(f(x)) \\ \lim_{t \to 0} \frac{f(x)^t-1}{t} = \ln(f(x))\]

Answer 6

OS lags too much for me these days to use it fast enough lol. XD I keep having to refresh to use the Post button.

Answer 7

since t is the variable, f(x) is a constant, say 'a'
so this gets converted into a standard limit....
which can be proved even by L'Hopital

Answer 8

same here :| the replies are dancing on my end lol
anyway to work this without LH or derivatives ? cuz in the previous post we wanted to work it without LH

Answer 9

Does this "prove" the limit though?

Answer 10

idk it looks like a legit proof to me if we believe in derivatives ?

Answer 11

I didn't really explain my version fairly well since I skipped steps that seem crucial. I am basically saying I'm using the chain rule for the right hand side of the second line of my "proof" since that's all it really is.

Answer 12

@SithsAndGiggles we wanted to work this without LH/derivatives in the previous post, i am still trying to cook up a proof using real analysis stuff like epsilon/delta, cauchy or opensets.. but no significant progress as such yet

Answer 13

This limit does not depend on x, so you can call a=f(x).
Let h(t) = a^t, then h'(t) = a^t ln(a)
the limit is h'(0)= ln(a)=ln(f(x))

Answer 14

I'm not sure about this myself, but does the squeeze theorem "prove" a limit exists, or is it just a way of computing a limit?

Answer 15

Wolframalpha knows that too
http://www.wolframalpha.com/input/?i=limit+%28+f%28x%29%5Et+-1%29%2Ft+when+t+goes+to+zero

Answer 16

You are just reproving some known fact
http://www.wyzant.com/resources/lessons/math/calculus/derivative_proofs/a_to_the_x

Answer 17

I see other ways to work this only complicate things ;) thanks everyone !

Answer 18

nice nice , was late to this lol

Answer 19

gimme an analysis proof after your exams @Marki

Answer 20

u wanna question , or wanna me to answer them B-) ?

Answer 21

or you wanna how to prove this with analysis

Answer 22

yes using analysis
see
http://math.stackexchange.com/questions/493849/evaluating-lim-t-to-0-left-int-01bxa1-xt-mathrm-dx-right1-t

Answer 23

look at robjohn's reply

Answer 24

i wonder what condition on f(x) , does it work when f(something )=0
it should be f(x)>0 , right ?

Answer 25

Um, what in the world is this? You're not making no sense.

Answer 26

oh so u have other sense about this :P *doubt*