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?
f(x) = 10x + e−x; g(x) =1 divided by the quantity 5 times x
f(x) = x2; g(x) = e−4x
f(x) =(lnx)3; g(x) =1 divided by x
f(x) =square root of x; g(x) = e−x

Let's check these folks using LH, or L'Hopital rule, since they're indeterminate types.



on this pair, the denominator is moving faster than the numerator, and thus, yielding a much larger denominator on every iteration, moving towards a limit of 0.



on this pair, the denominator, again, is moving faster, since ln() results in an exponent for the constant "e", and thus is a much smaller then the denominator on every iteration, thus, the denominator gets much larger and moving towards 0 as limit as well.



on this pair, the numerator is just a constant, thus is static, and the denominator is moving onwards on every iteration, thus moving also again towards 0.

now, let's take a peek of the 1st pair for f(x) and g(x)