Expert: Paul Klarreich - 8/21/2010

Question
Hi, I'm having trouble solving this problem on limits on my calc packet.

lim (e^2x - e^4)/(x-2)
x->2

Thanks.

Answer
I think this is one of those "do you recognize the definition of the derivative?"  exercises:

Definition of f'(2):
                 f(h) - f(2)
f'(2) = lim[h->2] ------------
                    h - 2

Now compare this with your thing:

         e^2x - e^4
lim[x->2] ------------
           (x-2)

which is the same as (recall that the 'x' here is a dummy):

         e^2h - e^4
lim[h->2] ------------
           (h-2)

and deduce (authors love that word, don't they?) that, if we make

f(x) = e^2x,

then

f(2) = e^4, and your expression is:

         e^2h - e^4
lim[h->2] ----------- = f'(2)
           (h-2)

Now if  f(x) = e^2x, then f'(x) = 2 e^2x,

and you can finish up.