Verified answer

lim = 0

lim x-> inf e^-2x = lim x-> inf a million/e^2x = 0 because of the fact e^2x is an fairly huge huge type. because of the fact cos oscillates between -a million and + a million cos(x) / e^2x = 0 while x -> inf lim as x->infinity of (e^-2x cos x) is a classic damped oscillation

Recognize that while cos(inf) has no definite answer, it is still bounded between -1 and 1, so:

e^(-2x) * cos(x) =>

cos(x) / e^(2x)

cos(inf) / e^(2 * inf) => L

-1 / e^inf </= L </= 1 / e^inf

0 </= L </= 0

So, L must equal 0