Why can't we just say that $\infty-\infty$ equals zero? Let be $\lim\limits_{x\to \infty}x=A$  and   $\lim\limits_{y\to
 \infty}y=B$.


Can be $A-B=0$?

If the answer is "no" , why?

And my other example:


  $\displaystyle\int_{-\infty}^{\infty} \dfrac{x}{1+x^2}dx$


Why does this not equal zero?

I know this is an improper integral, and I know how solve this. But I don't think this makes sense.

We have a positive area A and a negative area B (with $A=-B$).

Why can't we just say that 


  $\displaystyle\int_{-\infty}^{\infty} \dfrac{x}{1+x^2}dx=0$


And my last example:


  $\displaystyle\sum_{n=0}^\infty n^2=A, \quad \displaystyle\sum_{b=0}^\infty b^2=B$


Here can we say that $A-B=0$? If "no", then why?
Hint. Two examples. 

Observe that, as $x \to \infty$,
$$
(x^2+x)-x^2=x \to \infty
$$ whereas, as $x \to \infty$,
$$
\left(x+\frac1x\right)-x=\frac1x \to 0.
$$ In each case you have an indeterminate form $\infty-\infty$.
The simplest answer to your question, without either sequences or limits, is that there exist distinct infinite numbers. The most commonly used extension of the real numbers to an ordered field is the field of hyperreals. If $H$ is an infinite hyperreal, then so is $2H$ or for example $H^2$ or $\sqrt{H}$, not to speak of $H+1$ :-)