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more specifically, in my homework i have a question where,

lim f(x) converges

x->inf

lim g(x) diverges

x->inf

and i am asked about convergence of lim u(x) when

1) u(x)=f(x)+g(x)

2) u(x)=f(x)*g(x)

for 1) i say

lim f(x)+g(x)= lim f(x) + lim g(x)====> diverges

for 2) I am not sure, but i think

if f(X) converges to any number, K*inf=inf and u(x) diverges

if f(X) converges to 0 , then 0*inf is undefined and u(x) diverges

1st of all i am not 100% that i am right, second of all how do i prove this MATHEMATICALLY?