The limit of $\frac{3^x+4^x}{2^x + 5^x}$ as x approaches infinity

Question

How do I do this problem? The x is in the exponents.

Answer 1

Factor out a $5^x$ from the numerator and denominator: $$ \frac{3^x+4^x}{2^x+5^x}=\frac{(3/5)^x+(4/5)^x}{(2/5)^x+1} $$ Now what happens as $x\to\infty$?

Answer 2

Hint : Divide numerator and denominator by $5^x$