1. The problem statement, all variables and given/known data Prove or disprove: You have the functions g:R->R and f:R->R: 1) If the limit of g at x0 is infinity and the limit of f*g (multiplication) at x0 is also infinity then there's some neibourhood of x0 where f(x)>0 for every x in the neibourhood. 2)f and g are defined only in [0,infinity) and and L is in R: If the limit of f*g at infinity is L and the limit of f at infinity is infinity then the limit of g at infinity exists. 2. Relevant equations 3. The attempt at a solution 1) False: g(x) = 1/(x^2) and f(x) = 1 for all x=/=0 and f(x) = -1 for x=0. Is that right? It seemed to trivial. 2) True: For every E (epsilon) >0 we can find M>0 so that ME>|L| => ME-|L| > 0. Also, we can find N_1>0 so that for all x>N_1 f(x)>M => |f(x)| > M. Also, we can find N_2>0 so that for all x>N_2 |f(x)g(x)-L|< ME-|L| . So if N>max(N_1,N_2) then: ME-|L| < |f(x)g(x)-L|<= |f(x)g(x)|-|L| < M|g(x)|-|L| => |g(x)| < E and so we found that the limit of g at infinity is 0. Is that right? It seems weird that in the question they only proved that the limit exists and I found that it's always 0. Thanks.