Can Math Prove Lim f(x)*g(x) Convergence/Divergence?

[Dell]

if i have a lim of f(x)*g(x) can i say it is limf(x)*limg(x) like i could if i had lim of f(x)+g(x),
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?

 

[tiny-tim]

lim f(x) converges
x->inf

lim g(x) diverges
x->inf

and i am asked about convergence of lim u(x) when
…
2) u(x)=f(x)*g(x)

Hi Dell!

(have an infinity: ∞ )

Hint: x -> ∞ for f(x) = 1/x², g(x) = x³ ?

x -> ∞ for f(x) = 1/x², g(x) = x ?

 

[Dell]

in that case it really does diverge, but can i do that, just take any 2 functions and see what happens with them?? surely that is just one example and not enough to prove anything

 

[Dell]

o i see what you are saying, you are DISPROVING what i said,
so what does this mean? tha nothing can be said about fx*gx?? was i right about fx+gx??

 

[tiny-tim]

o i see what you are saying, you are DISPROVING what i said,
so what does this mean? tha nothing can be said about fx*gx?? was i right about fx+gx??

Yes and yes