derivative of lim x->infinite (2^x+5^x)^(1/x)?

Question

I do it till [2^xln(2)+5^xln(5)]/(2^x+5^x) and then I can't simplify anymore. Help me out?

Answer 1

We'll first find the limit, and I'll let you continue with the derivative.

Let
Y=Lim x->∞ (2^x+5^x)^(1/x)
=Lim x->∞ ((5*2/5)^x+5^x)^(1/x)
=Lim x->∞ ((5^x)(2/5)^x+5^x)^(1/x) [law of exponents]
=Lim x->∞ ((5^x)((2/5)^x+1)^(1/x) [factorize]

Take (natural) log on both sides:
log(Y)=Lim x->∞ (1/x)(xlog(5)+log(1+(2/5)^x)
Take limit x->∞,
(2/5)^x -> 0, hence log(1)=0
so
log(Y)=log(5)
=> Y=5=Lim x->∞ (2^x+5^x)^(1/x)

Answer 2

I don't get what you did after log(Y)=Lim x->∞ (1/x)(xlog(5)+log(1+(2/5)^x) . Why is (2/5)^x -> 0, and hence log(1)=0??