Editing Indeterminate form (section)

=== Indeterminate form 0/0 ===
{{Redirect|0/0|the symbol|Percent sign|0 divided by 0|Division by zero}}
<gallery>
File:Indeterminate form - x over x.gif|Fig. 1: {{var|y}} = {{sfrac|{{var|x}}|{{var|x}}}}
File:Indeterminate form - x2 over x.gif|Fig. 2: {{var|y}} = {{sfrac|{{var|x}}{{sup|2}}|{{var|x}}}}
File:Indeterminate form - sin x over x close.gif|Fig. 3: {{var|y}} = {{sfrac|sin&nbsp;{{var|x}}|{{var|x}}}}
File:Indeterminate form - complicated.gif|Fig. 4: {{var|y}} = {{sfrac|x &minus; 49|{{radic|x}} &minus; 7}} (for {{var|x}} = 49)
File:Indeterminate form - 2x over x.gif|Fig. 5: {{var|y}} = {{sfrac|{{var|a}}{{var|x}}|{{var|x}}}} where {{var|a}} = 2
File:Indeterminate form - x over x3.gif|Fig. 6: {{var|y}} = {{sfrac|{{var|x}}|{{var|x}}{{sup|3}}}}
</gallery>
The indeterminate form <math>0/0</math> is particularly common in [[calculus]], because it often arises in the evaluation of [[derivative]]s using their definition in terms of limit.

As mentioned above,
{{block indent|<math> \lim_{x \to 0} \frac{x}{x} = 1, \qquad </math> (see fig. 1)}}
while
{{block indent|<math> \lim_{x \to 0} \frac{x^{2}}{x} = 0,  \qquad </math> (see fig. 2)}}
This is enough to show that <math>0/0</math> is an indeterminate form. Other examples with this indeterminate form include
{{block indent|<math> \lim_{x \to 0} \frac{\sin(x)}{x} = 1, \qquad </math> (see fig. 3)}}
and
{{block indent|<math> \lim_{x \to 49} \frac{x - 49}{\sqrt{x}\, - 7} = 14, \qquad </math> (see fig. 4)}}
Direct substitution of the number that ''<math>x</math>'' approaches into any of these expressions shows that these are examples correspond to the indeterminate form <math>0/0</math>, but these limits can assume many different values. Any desired value <math>a</math> can be obtained for this indeterminate form as follows:
{{block indent|<math> \lim_{x \to 0} \frac{ax}{x} = a . \qquad </math> (see fig. 5)}}
The value <math>\infty</math> can also be obtained (in the sense of divergence to infinity):
{{block indent|<math> \lim_{x \to 0} \frac{x}{x^3} = \infty . \qquad </math> (see fig. 6)}}