Re: Limits

In article
<8fc52279-6af8-464f-b7aa-c2bd46960df1@xxxxxxxxxxxxxxxxxxxxxxxxxx>,
"Dave L. Renfro" <renfr1dl@xxxxxxxxx> wrote:

6. limit as x --> 0 of (1 + ax)^(1/x) is equal to exp(a)

7. limit as x --> 0 of (x + a^x)^(1/x) is equal to ae

8. limit as x --> oo of (1 + ax)^(1/x) is equal to 1 {{a > 0}}

9. limit as x --> 0 of [ax + exp(bx)] ^ (c/x) is equal to exp[c(a+b)]

11. limit as x --> 0 of [(a^x + b^x + c^x)/3] ^ (3/x) is equal to abc

19. limit as x--> 0 of [1 + (sin 2x)] ^ (1/x) is equal to exp(2)

23. limit as x --> 0 of [(sin x)/x + x] ^ (1/x) is equal to e

All of these can be handled with the following result: If f(0) = 1 and
f'(0) exists, then f(x)^(1/x) -> e^(f'(0)) as x -> 0. Proof: Taking
logs, and then using the definition of the derivative, we have
ln(f(x))/x = [ln(f(x)) - ln(f(0))]/x -> (ln(f(x)))'(0) = f'(0)/f(0) =
f'(0). Now exponentiate.
.