式展開: lim x→π/2 cot(x) を求める

$lim_{x \to \frac{π}{2}} \cot (x)$ を求める

高階の微分を求めておく

準備として

\csc (x)

の微分を求める．

\begin{array}{rcl} \frac{d}{d x} \csc (x) & = & \frac{d}{d x} \frac{1}{\sin (x)} \\ = & \sin^{- 1} (x) \\ = & - \sin^{- 2} (x) (\frac{d}{d x} \sin (x)) \dots u = \sin (x), f = u^{- 1}, \frac{d f}{d x} = \frac{d f}{d u} \frac{d u}{d x} = - u^{- 2} \frac{d u}{d x} \\ = & - \sin^{- 2} (x) (\cos (x)) \\ = & - \frac{\cos (x)}{\sin^{2} (x)} \\ = & - \frac{1}{\sin (x)} \frac{\cos (x)}{\sin (x)} \\ = & - \csc (x) \cot (x) \end{array}

一階から順に求めておく．

\begin{array}{rcl} \frac{d}{d x} \cot (x) & = & \frac{d}{d x} \frac{\cos (x)}{\sin (x)} \\ = & \frac{(\frac{d}{d x} \cos (x)) \sin (x) - \cos (x) (\frac{d}{d x} \sin (x))}{\sin^{2} (x)} \dots {(\frac{u}{v})}^{'} = \frac{u^{'} v - u v^{'}}{v^{2}} \\ \dots {(\frac{u}{v})}^{'} = {(u v^{- 1})}^{'} = u {(v^{- 1})}^{'} + (u^{'}) v^{- 1} = u (- v^{- 2} v^{'}) + (u^{'}) v^{- 1} \cdot v v^{- 1} = v^{- 2} (u^{'} v - u v^{'}) = \frac{u^{'} v - u v^{'}}{v^{2}} \\ = & \frac{(- \sin (x)) \sin (x) - \cos (x) (\cos (x))}{\sin^{2} (x)} \\ = & - \frac{\sin^{2} (x) + \cos^{2} (x)}{\sin^{2} (x)} \\ = & - \frac{1}{\sin^{2} (x)} \\ = & - \csc^{2} (x) \\ \frac{d^{2}}{d x^{2}} \cot (x) & = & \frac{d}{d x} (- \csc^{2} (x)) \\ = & - 2 \csc (x) \frac{d}{d x} \csc (x) \dots u = \csc (x), f = - u^{2}, \frac{d f}{d x} = \frac{d f}{d u} \frac{d u}{d x} = - 2 u \frac{d u}{d x} \\ = & - 2 \csc (x) (- \csc (x) \cot (x)) \dots 準 備 参 照 \\ = & 2 \csc^{2} (x) \cot (x) \dots \frac{d}{d x} \csc^{2} (x) = - 2 \csc^{2} (x) \cot (x) で も あ る (後 で 使 う) ． \\ \frac{d^{3}}{d x^{3}} \cot (x) & = & \frac{d}{d x} (2 \csc^{2} (x) \cot (x)) \\ = & 2 \frac{d}{d x} (\csc^{2} (x) \cot (x)) \\ = & 2 {(\frac{d}{d x} \csc^{2} (x)) \cot (x) + \csc^{2} (x) (\frac{d}{d x} \cot (x))} \dots (u v)^{'} = u^{'} v + u v^{'} \\ = & 2 {(- 2 \csc^{2} (x) \cot (x)) \cot (x) + \csc^{2} (x) (- \csc^{2} (x))} \dots \frac{d}{d x} \csc^{2} (x) = - 2 \csc^{2} (x) \cot (x), \frac{d}{d x} \cot (x) = - \csc^{2} (x) \\ = & - 2 (2 \csc^{2} (x) \cot^{2} (x) + \csc^{4} (x)) \\ \frac{d^{4}}{d x^{4}} \cot (x) & = & \frac{d}{d x} (- 2 (2 \csc^{2} (x) \cot^{2} (x) + \csc^{4} (x))) \\ = & - 2 [\frac{d}{d x} (2 \csc^{2} (x) \cot^{2} (x) + \csc^{4} (x))] \\ = & - 2 [\frac{d}{d x} (2 \csc^{2} (x) \cot^{2} (x)) + \frac{d}{d x} (\csc^{4} (x))] \dots (u + v)^{'} = u^{'} + v^{'} \\ = & - 2 [2 {(\frac{d}{d x} \csc^{2} (x)) \cot^{2} (x) + \csc^{2} (x) (\frac{d}{d x} \cot^{2} (x))} + 4 \csc^{3} (x) (- \csc (x) \cot (x))] \dots (u v)^{'} = u^{'} v + u v^{'}, u = \csc (x), f = u^{4}, \frac{d f}{d x} = \frac{d f}{d u} \frac{d u}{d x} = 4 u^{3} \frac{d u}{d x}, \frac{d}{d x} \csc (x) = - \csc (x) \cot (x) \\ = & - 2 [2 {(- 2 \csc^{2} (x) \cot (x)) \cot^{2} (x) + \csc^{2} (x) (2 \cot (x) (- \csc^{2} (x)))} - 4 \csc^{4} (x) \cot (x)] \dots \frac{d}{d x} \csc^{2} (x) = - 2 \csc^{2} (x) \cot (x), u = \cot (x), f = u^{2}, \frac{d f}{d x} = \frac{d f}{d u} \frac{d u}{d x} = 2 u \frac{d u}{d x}, \frac{d}{d x} \cot (x) = - \csc^{2} (x) \\ = & - 2 [2 {- 2 \csc^{2} (x) \cot^{3} (x) - 2 \csc^{4} (x) \cot (x)} - 4 \csc^{4} (x) \cot (x)] \\ = & - 2 [- 4 \csc^{2} (x) \cot^{3} (x) - 4 \csc^{4} (x) \cot (x) - 4 \csc^{4} (x) \cot (x)] \\ = & - 2 [- 4 \csc^{2} (x) \cot^{3} (x) - 8 \csc^{4} (x) \cot (x)] \\ = & 8 \csc^{2} (x) \cot (x) (\cot^{2} (x) + 2 \csc^{2} (x)) \\ \frac{d^{5}}{d x^{5}} \cot (x) & = & \frac{d}{d x} {8 \csc^{2} (x) \cot (x) (\cot^{2} (x) + 2 \csc^{2} (x))} \\ = & 4 \frac{d}{d x} {2 \csc^{2} (x) \cot (x) (\cot^{2} (x) + 2 \csc^{2} (x))} \\ = & 4 {(\frac{d}{d x} 2 \csc^{2} (x) \cot (x)) (\cot^{2} (x) + 2 \csc^{2} (x)) + 2 \csc^{2} (x) \cot (x) (\frac{d}{d x} (\cot^{2} (x) + 2 \csc^{2} (x)))} \\ = & 4 [{- 2 (2 \csc^{2} (x) \cot^{2} (x) + \csc^{4} (x))} (\cot^{2} (x) + 2 \csc^{2} (x)) + 2 \csc^{2} (x) \cot (x) (\frac{d}{d x} \cot^{2} (x) + 2 \frac{d}{d x} \csc^{2} (x))] \\ = & 4 [(- 4 \csc^{2} (x) \cot^{2} (x) - 2 \csc^{4} (x)) (\cot^{2} (x) + 2 \csc^{2} (x)) + 2 \csc^{2} (x) \cot (x) {(- 2 \csc^{2} (x) \cot (x)) + 2 (- 2 \csc^{2} (x) \cot (x))}] \\ = & 4 {(- 4 \csc^{2} (x) \cot^{2} (x) (\cot^{2} (x) + 2 \csc^{2} (x)) - 2 \csc^{4} (x) (\cot^{2} (x) + 2 \csc^{2} (x))) + 2 \csc^{2} (x) \cot (x) \cdot - 2 \csc^{2} (x) \cot (x) + 2 \csc^{2} (x) \cot (x) \cdot 2 (- 2 \csc^{2} (x) \cot (x))} \\ = & 4 (- 4 \csc^{2} (x) \cot^{4} (x) - 8 \csc^{4} (x) \cot^{2} (x) - 2 \csc^{4} (x) \cot^{2} (x) - 4 \csc^{6} (x) - 4 \csc^{4} (x) \cot^{2} (x) - 8 \csc^{4} (x) \cot^{2} (x)) \\ = & 4 (- 4 \csc^{2} (x) \cot^{4} (x) - 4 \csc^{6} (x) - 22 \csc^{4} (x) \cot^{2} (x)) \\ = & - 8 (2 \csc^{6} (x) + 2 \csc^{2} (x) \cot^{4} (x) + 11 \csc^{4} (x) \cot^{2} (x)) \end{array}

$x = \frac{π}{2}$ でのテーラー展開を求めておく

\begin{array}{rcl} \cot (x) & = & \frac{1}{0!} [{\frac{d^{0}}{d x^{0}} \cot (x) |}_{x = \frac{π}{2}}] {(x - \frac{π}{2})}^{0} \\ + \frac{1}{1!} [{\frac{d^{1}}{d x^{1}} \cot (x) |}_{x = \frac{π}{2}}] {(x - \frac{π}{2})}^{1} \\ + \frac{1}{2!} [{\frac{d^{2}}{d x^{2}} \cot (x) |}_{x = \frac{π}{2}}] {(x - \frac{π}{2})}^{2} \\ + \frac{1}{3!} [{\frac{d^{3}}{d x^{3}} \cot (x) |}_{x = \frac{π}{2}}] {(x - \frac{π}{2})}^{3} \\ + \frac{1}{4!} [{\frac{d^{4}}{d x^{4}} \cot (x) |}_{x = \frac{π}{2}}] {(x - \frac{π}{2})}^{4} \\ + \frac{1}{5!} [{\frac{d^{5}}{d x^{5}} \cot (x) |}_{x = \frac{π}{2}}] {(x - \frac{π}{2})}^{5} \\ + \dots \\ = & \frac{1}{1} [\cot (\frac{π}{2})] {(x - \frac{π}{2})}^{0} \\ + \frac{1}{1} [- \csc^{2} (\frac{π}{2})] {(x - \frac{π}{2})}^{1} \\ + \frac{1}{2} [2 \cot (\frac{π}{2}) \csc^{2} (\frac{π}{2})] {(x - \frac{π}{2})}^{2} \\ + \frac{1}{6} [- 2 (2 \csc^{2} (\frac{π}{2}) \cot^{2} (\frac{π}{2}) + \csc^{4} (\frac{π}{2}))] {(x - \frac{π}{2})}^{3} \\ + \frac{1}{24} [8 \csc^{2} (\frac{π}{2}) \cot (\frac{π}{2}) (\cot^{2} (\frac{π}{2}) + 2 \csc^{2} (\frac{π}{2}))] {(x - \frac{π}{2})}^{4} \\ + \frac{1}{120} [- 8 (2 \csc^{6} (\frac{π}{2}) + 2 \csc^{2} (\frac{π}{2}) \cot^{4} (\frac{π}{2}) + 11 \csc^{4} (\frac{π}{2}) \cot^{2} (\frac{π}{2}))] {(x - \frac{π}{2})}^{5} \\ + \dots \\ = & [0] \cdot 1 \\ + [- 1 \cdot 1^{2}] (x - \frac{π}{2}) \\ + \frac{1}{2} [2 \cdot 0 \cdot 1^{2}] {(x - \frac{π}{2})}^{2} \\ + \frac{1}{6} [- 2 (2 \cdot 1^{2} \cdot 0^{2} + 1^{4})] {(x - \frac{π}{2})}^{3} \\ + \frac{1}{24} [8 \cdot 1^{2} \cdot 0 (0^{2} + 2 \cdot 1^{2})] {(x - \frac{π}{2})}^{4} \\ + \frac{1}{120} [- 8 (2 \cdot 1^{6} + 2 \cdot 1^{2} \cdot 0^{4} + 11 \cdot 1^{4} \cdot 0^{2})] {(x - \frac{π}{2})}^{5} \\ + \dots \\ = & - (x - \frac{π}{2}) - \frac{2}{6} {(x - \frac{π}{2})}^{3} - \frac{16}{120} {(x - \frac{π}{2})}^{5} + \dots \\ = & - (x - \frac{π}{2}) - \frac{1}{3} {(x - \frac{π}{2})}^{3} - \frac{2}{15} {(x - \frac{π}{2})}^{5} + \dots \end{array}

$lim_{x \to \frac{π}{2}} \cot (x)$ を求める

\begin{array}{rcl} lim_{x \to \frac{π}{2}} \cot (x) & = & lim_{x \to \frac{π}{2}} [- (x - \frac{π}{2}) - \frac{1}{3} {(x - \frac{π}{2})}^{3} - \frac{2}{15} {(x - \frac{π}{2})}^{5} + \dots] \\ = & 0 \end{array}

式展開

lim x→π/2 cot(x) を求める

$lim_{x \to \frac{π}{2}} \cot (x)$ を求める

高階の微分を求めておく

$x = \frac{π}{2}$ でのテーラー展開を求めておく

$lim_{x \to \frac{π}{2}} \cot (x)$ を求める

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式展開

lim x→π/2 cot(x) を求める

limx→π2cot⁡(x)を求める

高階の微分を求めておく

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limx→π2cot⁡(x)を求める

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$lim_{x \to \frac{π}{2}} \cot (x)$ を求める

$x = \frac{π}{2}$ でのテーラー展開を求めておく

$lim_{x \to \frac{π}{2}} \cot (x)$ を求める