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## Main Question or Discussion Point

Hey, I am having some difficulties. So it's my understanding that the function e

f'(x) = lim h-->0 (f(x+h)-f(x))/h = lim h-->0 (a

So the goal is, if we can find a value a that makes (lim h--> 0 (a

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

lim h-->0 (h*a

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

^{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 derivativef'(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 getlim h-->0 (h*a

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

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