Question

For which pair of functions f(x) and g(x) below will the lim(x->infinity) f(x)g(x)≠0

a)f(x) = 10x + e^-x; g(x) = (1/5x)
b)f(x) = x^2; g(x) = e^-4x
c)f(x) = (Lnx)^3; g(x) = 1/x
d)f(x) = √x; g(x) = e^-x

Answer 1

exponentials grow and decay faster than polynomials. So, b,d -> 0

logs grow more slowly than polynomials, so c -> 0

a does not go to zero, since

f(x)g(x) = 2x^2 + x/5 e^-x -> 2x^2