Can I subtract infinity from infinity?

Question

I was stuck when solving a problem on limits. It was like----> $\lim_{x\to\infty} (x-x)$. What should I do now?

Answer 1

The answer is 1 as first you simplify limit expression and then plug in value to check the limit if it's not indeterminate. So the limit is$100

Answer 2

First of all: you cannot just subtract infinity from infinity. Infinity is not a real number so you can't simply use the basic operations as you're used to do with (real) real numbers.

However, in the context of limits, there are things you can do. As pointed out, the limit in your question is $0$ because $x-x=0$ for all $x$, so the 'calculation' would be something like: $$\lim_{x \to +\infty} (x - x) = \lim_{x \to +\infty} 0 = \color{green}{0}$$ When you would not do that second step of simplification but rather take the limit of both terms, you get a so called indeterminate form $\infty-\infty$. To increase your understanding of what that means, it's useful to compare your limit with, for example, the following two: $$\lim_{x \to +\infty} (2x - x) \quad \mbox{and} \quad \lim_{x \to +\infty} (x - 2x)$$ When you, again, just 'fill in' (take the limit of both terms separately), you get the same form $\infty-\infty$ but after simplifying, the limits turn out to be: $$\lim_{x \to +\infty} (2x - x) = \lim_{x \to +\infty} x = \color{blue}{+\infty} \quad \mbox{and} \quad \lim_{x \to +\infty} (x - 2x) = \lim_{x \to +\infty} -x = \color{red}{-\infty} $$ Where you found $\color{green}{0}$ for your limit, we now found $\color{blue}{+\infty}$ and $\color{red}{-\infty}$ for the two variants, all of which were an indeterminate $\infty-\infty$ at first. The fact that this can happen is precisely the reason to say that '$\infty-\infty$' is an indeterminate form. In fact, you can come up with examples where any real number $c$ is the result, starting from this indeterminate form.