Hmm. Let's see.
There is where you ended. At this point I would divide top and bottom by instead of using L'Hopital's Rule again.
(4^x + 7^x)^(1/x)
First I took the ln of (4^x + 7^x)^(1/x) to get (4^x + 7^x)/x and derived the top and bottom, so I got a function of
(4^x *ln4 + 7^x *ln7)/(4^x + 7^x)
and I took the derivative of the top and bottom again, but I could tell already that things were going nowhere :/
ohhhhhhhh i see now. Dividing the top and bottom by 7^x yields ((((4^x)*ln4) /7^x) + ln7)/((4^x/7^x)+1) and 4^x*ln4/7^x and 4^x / 7^x both go to zero since 7^x will grow at a faster rate than 4^x, so ln (4^x + 7^x)^(1/x) = e^(ln7) = 7
Thank you so much!
But real quick: Why did you have to divide by 7^x to get the right answer?