Indeterminate expressions

Here we will look closer at all indeterminate expressions. We will illustrate by examples that each indeterminate expression can yield any concievable answer, including any (positive) real number, zero, (minus) infinity, or the limit may not exist.

The examples will be based on the following facts. From Theory - Limits - Important examples we know that 1/n→0 and that the limit of (−1)ⁿ does not exists. On the other hand, we also know (see for instance here) that (−1)ⁿ/n→0, n + (−1)ⁿ→∞ and n(2 + (−1)ⁿ)→∞.

In the following examples, c stands for any real number unless noted otherwise.

The indeterminate expression ∞ − ∞:

These examples show that the expression ∞ − ∞ can indeed lead to any conclusion.

The indeterminate expression 0⋅∞:

Again, we see that the expression 0⋅∞ can be anything.

The indeterminate expression :

Now we can get anything between zero and infinity, included, and also DNE.

The indeterminate expression :

So we can get anything (real numbers, infinity, DNE) from .

The indeterminate expression 1/0: Here we have three possible outcomes:

The indeterminate expression 1^∞:

Here c was any real number, so we can make the expression 1^∞ go to any positive number, or zero, or infinity, or DNE.

The indeterminate expression 0⁰:

Here c was any real number, so again, 0⁰ may go to any positive number, or zero, or infinity, or DNE. The same is true about the last case below.

The indeterminate expression ∞⁰: