Aha, alright.
We have an indeterminate form 0/0. There are several approaches on how to get rid of it. Here's one.
First, let's divide everything by 9^x.
(4^x - 9^x)/(x(4^x + 9^x)) = ((4/9)^x - 1)/(x((4/9)^x + 1))
First, we notice that there's no problem with (4/9)^x + 1, so let's just ignore it for now.
((4/9)^x - 1)/x
Let f(x) = (4/9)^x. Then our limit is:
(f(x) - f(0))/(x - 0), x -> 0
We can recognize it as the definition of the derivative:
(f(x) - f(0))/(x - 0) -> f'(0), x -> 0
Let's find f'(0).
f'(x) = ln(4/9) (4/9)^x
f'(0) = ln(4/9)
Thus:
((4/9)^x - 1)/x -> ln(4/9), x -> 0
Finally, we brink back (4/9)^x + 1:
((4/9)^x - 1)/(x((4/9)^x + 1)) -> ln(4/9)/(1 + 1) = (1/2)ln(4/9), x -> 0
It's also convenient to notice that ln(4/9) = ln((2/3)^2) = 2ln(2/3). So:
((4/9)^x - 1)/(x((4/9)^x + 1)) -> ln(2/3), x -> 0
Thus, the limit is equal to ln(2/3).
#limits
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