What are the commonly encountered indeterminate forms in algebraic functions?

[Greg Bernhardt]

Definition/Summary

An algebraic function of a pair of numbers is an indeterminate form at a particular pair of values (which may include infinity) if the function does not tend to a unique limit at that pair of values.

For example, $\infty\ -\ \infty$ is an indeterminate form because the function $f(x - y)$ does not tend to a unique limit at the values $x\ =\ y\ =\ \infty$.

Equations

Commonly encountered indeterminate forms:
$$0/0,~ \infty/\infty,~0\cdot \infty,~0^0,~1^{\infty},~\infty ^0,~\infty-\infty$$

Commonly encountered forms which are not indeterminate:
$$1/0,~0/\infty,~\infty/0,~0^{\infty}$$

Extended explanation

Examples:

1. Consider the form $\infty-\infty$, which can be made from the limit as $x\rightarrow \infty$ of $x^2-x$, $x-x$ or $x-x^2$. In these three cases, we find the limits are $\infty$, $0$ and $-\infty$ respectively.

2. Consider the form $\infty /\infty$, which can be made from the limit as $x\rightarrow \infty$ of $x^2/x$, $x/x$ or $x/x^2$. In these three cases, we find the limits are $\infty$, $1$ and $0$ respectively.

The lack of uniqueness makes these forms indeterminate.

l'Hôpital's rule:

If $f(x)\ =\ g(x)/h(x)$ and $g(a)/h(a)$ is an indeterminate form, $0/0$ or $\infty/\infty$, at some value $a$, but $g'(a)/h'(a)$ is not, then l'Hôpital's rule can be used to find the limiting value $f(a)$:

$$\lim_{x \rightarrow a} \frac{g(x)}{h(x)} = \lim_{x \rightarrow a} \frac{g'(x)}{h'(x)}$$

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[fresh_42]

I would refrain from using indeterminate forms. The risk for mistakes is high and inaccuracies can barely be avoided. Otherwise have a look on https://www.physicsforums.com/threads/what-are-extended-real-numbers.762879/ and the links in post #2 therein.