What is $0 \times \infty$? My question is - I know, $0\times anything=0$ and $anything \times \infty=\infty$.
So,what is $0 \times \infty$?
I suppose it's $0$ but why not $\infty$?
If I say that area of an indefinitely long line is $\infty*0=0$,where am I wrong?
I know upto limits and basic derivatives.
Thanks.

Maybe i should start of with: what does $\infty$ mean to you? 
Heuristically: In real analysis we assign the "value" $\pm \infty$ to limits that do not converge to any real number and are not bounded. 
(I suggest looking up the definition of convergence to infinity, as this was very heuristic and a counter-example to what i tried to describe in plain words can easily be found).
Anyways, once the definitions have been established - we can play around and come up with some useful THMs (Often referred to as limit artithmetics). 
Let $\{a_n\} \xrightarrow {n\rightarrow \infty} \infty$ and $\{b_n\} \xrightarrow {n\rightarrow \infty} 0$. Then for any $A \in \mathbb R \setminus \{0\}$, $B \in \mathbb R$. Then:
$A \cdot a_n \xrightarrow {n\rightarrow \infty} \infty$ 
$B \cdot b_n \xrightarrow {n\rightarrow \infty} 0$
What about $ a_n \cdot b_n \xrightarrow {n\rightarrow \infty}  ?$ 
Well, you can see (there are already plenty of examples in this page) that you can't tell what is the limit. You can't even say that it exists.
However, with the above "rules of arithemetics" (and knowing the commutative property) 
$$\lim _{n\rightarrow \infty} b_n \cdot A \cdot a_n=A \cdot \lim _{n\rightarrow \infty} b_n  \cdot a_n$$
And that is the most we can say. (Note that this is only meaningful if the limit exists).
