1. which grows fastest?

Which of the following functions grow faster then e^x as x approaches infinity?

x^4
ln(x)
e^(-x)
3^x
.5*e^x

2. 3^x because it is e^(xln 3) while e^x is just e^(x*1) and ln 3 > 1.

3. Originally Posted by ThePerfectHacker
3^x because it is e^(xln 3) while e^x is just e^(x*1) and ln 3 > 1.
Another reason as to why...

$3^x > e^x$

Same exponent, but the left side has a larger base hence a larger outcome, and to connect this to TPH's explanation..

$ln|3|x > x$

$x \ne 0$ <- this follows because at zero the functions are equal

$ln|3| > 1$