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which has no definite meaning, considering that division by zero is not a meaningful operation in arithmetic.

NOTA BENE: To say "0/0" is an indeterminate form does not mean simply that "0/0" by itself does not represent any number. Rather it means that if f(x) and g(x) both approach 0 as x approaches some limit c, one lacks sufficient information to evaluate the limit

That limit could be any number, could be infinite or could fail to exist, depending on what functions f and g are.

1 List of indeterminate forms

All of the following are indeterminate forms.

2 More on 0/0

If f(x) and g(x) both approach 0 as x approaches some number, or x approaches ∞ or −∞, then

can approach any real number or ∞ or −∞, or fail to converge to any point on the extended real number line, depending on which functions f and g are; similar remarks are true of the other indeterminate forms displayed above. For example,

and

Direct substitution of the number that x approaches into either of these functions leads to the indeterminate form 0/0, but both limits actually exist and are 1 and 14 respectively.

The indeterminate nature of the form does not imply the limit does not exist. In many cases, algebraic elimination, L'Hôpital's rule, or other methods can be used to simplify the expression so the limit can be more easily evaluated.

3 Naive arguments to give indeterminate forms a meaning

There are many naive reasons which may be given for considering indeterminate forms to have some definite meaning (for example):

• Anything divided by itself is 1. Hence
• Anything to the power of 0 is 1. Hence

The above are true statements if they are qualified by exceptions. Below are the correct versions of these statements.

• Any nonzero number divided by itself is 1.
• Any nonzero number to the power of 0 is 1.

The symbol does not represent a number. It represents a limit only. As such the following statements are entirely meaningless:

• Anything multiplied by is hence
• Anything multiplied by 0 is 0, hence
• Anything divided by is 0 Hence

(There are defined concepts such as the surreal numbers and the ordinals where operations on infinite objects are well defined.)

4 Logical circularity

In some cases it may constitute circular reasoning to use L'Hopital's rule to evaluate such limits as

If one uses the evaluation of the limit above for the purpose of proving that

(d/dx) xⁿ = nxⁿ⁻¹

and one uses L'Hopital's rule and the fact that

(d/dx) xⁿ = nxⁿ⁻¹

in the evaluation of the limit, then one's reasoning is circular and therefore fallacious.