What is infinity minus infinity?

Steps to Solve

1. Recognize the Indeterminate Form

   When we see $\infty - \infty$, it's not immediately clear what the result should be. Unlike regular arithmetic, infinity doesn't follow the same rules. Subtracting two infinitely large quantities doesn't necessarily result in zero or any other specific value.

2. Understand Indeterminate Forms

   In calculus and analysis, $\infty - \infty$ is known as an indeterminate form. This means that the limit of an expression that takes this form cannot be determined solely from the limits of the individual terms. You need more information about how the infinities are approached.

3. Consider Examples

   To illustrate why $\infty - \infty$ is indeterminate, consider a few examples:

   • Example 1: $\lim_{x \to \infty} (x^2 - x) = \infty$ Here, both $x^2$ and $x$ approach infinity as $x$ goes to infinity. However, $x^2$ grows much faster than $x$, so their difference tends to infinity.

   • Example 2: $\lim_{x \to \infty} (x - x^2) = -\infty$ Again, both $x$ and $x^2$ approach infinity. But now $x^2$ is subtracted from $x$, and since $x^2$ grows faster, the result tends to negative infinity.

   • Example 3: $\lim_{x \to \infty} (x - x) = 0$ In this trivial case, the difference is always zero, regardless of how large $x$ becomes.

   • Example 4: $\lim_{x \to \infty} (x + \sin(x) - x) = \lim_{x \to \infty} \sin(x)$. Here the limit does not exist since $\sin(x)$ oscillates between -1 and 1.

4. Conclusion: Indeterminate

   Since the result of $\infty - \infty$ can vary depending on the specific expressions involved, it is considered an indeterminate form. It does not have a single, well-defined value.