Editing Limit of a function (section)

===L'Hôpital's rule===
{{Main|l'Hôpital's rule}}
This rule uses [[derivative]]s to find limits of [[indeterminate forms]] {{math|0/0}} or {{math|±∞/∞}}, and only applies to such cases.  Other indeterminate forms may be manipulated into this form.  Given two functions {{math|''f''(''x'')}} and {{math|''g''(''x'')}}, defined over an [[open interval]] {{mvar|I}} containing the desired limit point {{mvar|c}}, then if:
# <math>\lim_{x \to c}f(x)=\lim_{x \to c}g(x)=0,</math>  or <math>\lim_{x \to c}f(x)=\pm\lim_{x \to c}g(x) = \pm\infty,</math> and
# <math>f</math> and <math>g</math> are differentiable over <math>I \setminus \{c\},</math> and
# <math>g'(x)\neq 0</math> for all <math> x \in I \setminus \{c\},</math> and
# <math>\lim_{x\to c}\tfrac{f'(x)}{g'(x)}</math>  exists,
then:
<math display=block>\lim_{x \to c} \frac{f(x)}{g(x)} = \lim_{x \to c} \frac{f'(x)}{g'(x)}.</math>

Normally, the first condition is the most important one.

For example:
<math>\lim_{x \to 0} \frac{\sin (2x)}{\sin (3x)} =
\lim_{x \to 0} \frac{2 \cos (2x)}{3 \cos (3x)} =
\frac{2 \sdot 1}{3 \sdot 1} =
\frac{2}{3}.</math>