Please help me on calculus: theorem

Question

For which pair of functions f(x) and g(x) below will the  ?

Options:

1.  f(x) = 10x + e^-x; g(x) = 
2.  f(x) = x²; g(x) = e^-4x
3.  f(x) =(Lnx)³; g(x) = 
4.  f(x) =  ; g(x) = e^-x

Answer 1

A product of 2 functions will go to zero if either function goes to zero. Therefore, for the product to NOT go to zero, neither function can go to zero.

So evaluate each function separately: The limit as x--->infinity:

1-f: depends on the sign of infinity, but cannot go to zero

1-g: goes to zero

2-f: goes to infinity

2-g: depends on the sign of infinity. for negative infinity, goes to zero

3-f: goes to infinity

3-g: goes to zero

4-f: goes to infinity

4-g: depends on the sign of infinity