What is the limit of this function as x approaches infinity?

Question

2^x + x^3 / x^2 + 3^x

Answer 1

use an intuitive approach

the exponential expression of 2^x and 3^x become larger at a much faster rate than do x^2 and x^3

so as x --> ∞

the expression is dominated by (2^x + ?)/(? + 3^x)
--> 2^x/3^x

now as x --->∞
2^x/3^x --- 0

so limit (2^x+x^3)/(x^2 + 3^x) as x --> ∞
= 0

test it, let x = 100

expression = (1.268 x10^30) + 1000000)/(10000 + 5.154x10^47) = 2.46 x 10^-18 (close to zero)