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Limits and L'Hôpital

A limit describes what a function approaches as xx approaches a certain point, even if the function may not be defined exactly there. L'Hôpital's rule gives a shortcut whenever we would otherwise get 0/00/0 or ∞/∞\infty/\infty.

lim⁡x→af(x)g(x)=lim⁡x→af′(x)g′(x)\lim_{x\to a}\frac{f(x)}{g(x)} = \lim_{x\to a}\frac{f'(x)}{g'(x)}L'Hôpital's rule (for 0/00/0 or ∞/∞\infty/\infty)

Symbols

aathe point the limit is taken towards

Example

lim⁡x→0sin⁡xx=1\displaystyle\lim_{x\to0}\frac{\sin x}{x} = 1 (the type 0/00/0).

L'Hôpital only applies to 0/00/0 or ∞/∞\infty/\infty - other forms must be rewritten first.
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Part of Calculus: Limits, series and complex numbers.