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.

$\infty$ is not a normal number and the rules of arithmetics only apply to normal numbers. The expression $0\times \infty$ is therefore not a sensible arithmetic expression that we can evaluate.
Instead, it is often used as a memonic when considering limits. If we have a sequence which grows without bounds (like $a_n=\{1,2,3,\ldots\}$) then we say $a_n\to \infty$ as $n\to \infty$ as the sequence grows without bounds. On the other hand for a sequence like $b_n = \{1 , \frac{1}{2},\frac{1}{3},\ldots\}$ we have $b_n\to 0$ as $n\to\infty$ as the terms approach $0$ as $n$ gets bigger and bigger.
Now if we have the product of two sequences $a_n\cdot b_n$ and ask what does $a_n\cdot b_n$ approach (if anything) when $n\to\infty$ then since $a_n\to\infty$ and $b_n\to 0$ we say that we have a limit on the form $0\cdot \infty$. In the case above this limit is just $1$ since $a_n\cdot b_n =\{1,1,1,\ldots\}$. Whenever you see $0\cdot \infty$ this is usually what is meant by it.
In this setting we can show that $0\cdot \infty$ is an indetermined form as it can be any number (or $\infty$) depending on the sequences we look at. We can find sequences $a_n\to\infty$ and $b_n\to 0$ such that $a_n\cdot b_n \to N$ for any real number $N$. Examples are given in the other answers.
