Why infinity minus infinity, 0 multiply infinity, infinity divided by infinity and 0/0 are not defined?

Question

Suppose that the arithmetic operation on extended complex plane are defined via arithmetic operations on the corresponding sequences, why $∞-∞，0*∞，∞/∞，0/0 $are not defined? Can anyone give me some example about that?

I know that ∞ + ∞ is undefined because if we have two sequences:

$1, 3, 3, 5, 5, 7, 7, 7, 9,....$ and

$-1,-2,-3,-4,-5,-6,-7,-8,....$

the sum of the above sequences will be: $0,1,0,1,0,1,0,1,...$

which is not converging or diverging. so ∞+∞ is meaningless.

Can I use it in $ ∞-∞，0*∞，∞/∞，0/0$?

Can anyone give me some example about that?

Answer 1

1. $[\infty-\infty]$

   • $\lim\limits_{n\to \infty} (n+1)^2-n^2=\infty$
   • $\lim\limits_{n\to \infty} (n^2+1)-n^2=1$

2. $[0\cdot\infty]$, $\left[\frac{\infty}{\infty}\right]$, $\left[\frac{0}{0}\right]$

   • $\lim\limits_{n\to \infty} \frac{1}{n}\cdot n^2=\lim\limits_{n\to \infty} \frac{n^2}{n}=\lim\limits_{n\to \infty} \frac{\frac{1}{n}}{\frac{1}{n^2}}=\infty$
   • $\lim\limits_{n\to \infty} \frac{1}{n}\cdot n=\lim\limits_{n\to \infty} \frac{n}{n}=\lim\limits_{n\to \infty} \frac{\frac{1}{n}}{\frac{1}{n}}=1$