Hey, I am having some difficulties. So it's my understanding that the function e(adsbygoogle = window.adsbygoogle || []).push({}); ^{x}comes around out of a desire to have a function whose derivative is equal to itself. Well we can show that if f(x)=a^{x}, a>0, then f'(x) is equal to a multiple of itself using the limit definition of the derivative

f'(x) = lim h-->0 (f(x+h)-f(x))/h = lim h-->0 (a^{x+h}- a^{x})/h = lim h-->0 a^{x}(a^{h}-1)/h = a^{x}( lim h-->0 (a^{h}-1)/h )

So the goal is, if we can find a value a that makes (lim h--> 0 (a^{h}-1)/h ) = 1, then f'(x) = f(x).

My only issue is that when I actually take this limit, I don't understand how it can be anything other than 0.

lim h-->0 (a^{h}-1)/h ) = 0/0 so if we apply L'hopitals, we get

lim h-->0 (h*a^{h-1})/(1) = [lim h--> 0 (h) * lim h-->0 (a^{h-1})]/lim h-->0 (1) = 0

Right? What am I doing wrong in evaluating this limit? I mean I know I"m doing something wrong I just can't figure out what it is

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Deriving the value of e

**Physics Forums | Science Articles, Homework Help, Discussion**