L'Hospital's Rule problem?

Question

lim (5^x - 3^x)/x
x-->0

If I plug in 0 for x, then I get 0/0.

So now I just have to use L'Hospital's rule. But I don't know how to get the derivitive of (5^x - 3^x)

The final answer is suppose to be: ln 5/3

Answer 1

the derivative of some constant a^x is

(a^x)*ln a

Answer 2

(a^x)' = ln a (a^x)

f(x) = 5^x - 3^x
f'(x) = ln 5(5^x) - ln 3(3^x)

lim (5^x - 3^x)/x
x=> 0

lim (ln 5(5^x) - ln 3(3^x))/1
x=>0

now plug in the 0
ln 5(5^0) - ln 3(3^0)= ln 5 - ln 3 = ln 5/3

Answer 3

lim (5^x - 3^x)/x
x-->0
lim (5^x ln5 - 3^x ln3)/1
x --> 0
= (ln5 -ln3)/1 = ln (5/3)