Question

CALC!! Please help me. For which pair of functions f(x) and g(x) below will the limit as x goes to infinity of the product of f of x and g of x does not equal 0? (I have attached a photo of the multiple choice answers and the question itself)

Answer 1

First limit:
$\lim_{x\to\infty}(10x+e^{-x})*\frac{1}{5x}$
$=\lim_{x\to\infty}(2+\frac{1}{5xe^{x}})$
$=2+0$
$=2$

Since we know that the first pair of functions does not approach 0, we do not have to check the others.