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0

The indeterminate form 0 / 0 {\displaystyle 0/0} is particularly common in calculus, because it often arises in the evaluation of derivatives using their definition in terms of limit.

As mentioned above,

while

This is enough to show that 0 / 0 {\displaystyle 0/0} is an indeterminate form. Other examples with this indeterminate form include

and

Direct substitution of the number that x {\displaystyle x} approaches into any of these expressions shows that these are examples correspond to the indeterminate form 0 / 0 {\displaystyle 0/0}, but these limits can assume many different values. Any desired value a {\displaystyle a} can be obtained for this indeterminate form as follows:

The value ∞ {\displaystyle \infty } can also be obtained (in the sense of divergence to infinity):