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)}<br /> = \lim_{x \rightarrow a} \frac{g&#039;(x)}{h&#039;(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.