Why can't we just say that $\infty-\infty$ equals zero?

Question

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?

Answer 1

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 $\infty-\infty$.

Answer 2

The form $\infty-\infty$ is what is known as an indeterminate form. This means as limits are concerned, the limit $\infty-\infty$ does not have a value.

Answer 3

if we have $$\lim_{n\to \infty}n^3=\infty$$ and $$\lim_{n \to \infty}n^2=\infty$$ then the difference goes to $$\infty$$

Answer 4

Infinite often is not an intuitive concept, but when a function goes to infinity its important to consider how it quickly grow. For example if you take $f(x)=3^x $ and $g(x)=x$ when $x \to +\infty, \space $ both functions go to infinity but $f$ is "faster" than $g$ so in this case $3^x-x=f_1(x)\to +\infty$ when $x\to +\infty$.

Answer 5

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 $H+1$ :-)