\(\lim_{x\rightarrow \frac{\pi}{2}} \cot{\left(x\right)}\)を求める
高階の微分を求めておく
準備として\(\csc{\left(x\right)}\)の微分を求める．
$$\begin{eqnarray}
\frac{\mathrm{d}}{\mathrm{d}x}\csc{\left(x\right)}
&=&\frac{\mathrm{d}}{\mathrm{d}x}\frac{1}{\sin{\left(x\right)}}
\\&=&\sin^{-1}{\left(x\right)}
\\&=&-\sin^{-2}{\left(x\right)}\left(\frac{\mathrm{d}}{\mathrm{d}x}\sin{\left(x\right)}\right)
\;\cdots\;u=\sin{\left(x\right)},f=u^{-1},\frac{\mathrm{d}f}{\mathrm{d}x}=\frac{\mathrm{d}f}{\mathrm{d}u}\frac{\mathrm{d}u}{\mathrm{d}x}=-u^{-2}\frac{\mathrm{d}u}{\mathrm{d}x}
\\&=&-\sin^{-2}{\left(x\right)}\left(\cos{\left(x\right)}\right)
\\&=&-\frac{\cos{\left(x\right)}}{\sin^2{\left(x\right)} }
\\&=&-\frac{1}{\sin{\left(x\right)}}\frac{\cos{\left(x\right)}}{\sin{\left(x\right)}}
\\&=&-\csc{\left(x\right)}\cot{\left(x\right)}
\end{eqnarray}$$
一階から順に求めておく．
$$\begin{eqnarray}
\frac{\mathrm{d}}{\mathrm{d}x}\cot{\left(x\right)}
&=&\frac{\mathrm{d}}{\mathrm{d}x}\frac{\cos{\left(x\right)}}{\sin{\left(x\right)}}
\\&=&\frac{
(\frac{\mathrm{d}}{\mathrm{d}x}\cos{\left(x\right)})\sin{\left(x\right)}
-\cos{\left(x\right)}(\frac{\mathrm{d}}{\mathrm{d}x}\sin{\left(x\right)})
}{\sin^2{\left(x\right)}}
\;\cdots\;\left(\frac{u}{v}\right)^\prime=\frac{u^\prime v-uv^\prime}{v^2}
\\&&\;\cdots\;\left(\frac{u}{v}\right)^\prime=\left(uv^{-1}\right)^\prime
=u\left(v^{-1}\right)^\prime+\left(u^\prime\right)v^{-1}
=u\left(-v^{-2}v^\prime\right)+ \left(u^\prime\right)v^{-1}\cdot vv^{-1}
=v^{-2}\left(u^\prime v-uv^\prime \right)
=\frac{u^\prime v-uv^\prime}{v^2}
\\&=&\frac{
(-\sin{\left(x\right)})\sin{\left(x\right)}
-\cos{\left(x\right)(\cos{\left(x\right)})}
}{\sin^2{\left(x\right)}}
\\&=&-\frac{
\sin^2{\left(x\right)}+\cos^2{\left(x\right)}
}{\sin^2{\left(x\right)}}
\\&=&-\frac{1}{\sin^2{\left(x\right)}}
\\&=&-\csc^2{\left(x\right)}
\\\;
\\\frac{\mathrm{d}^2}{\mathrm{d}x^2}\cot{\left(x\right)}
&=&\frac{\mathrm{d}}{\mathrm{d}x}\left(-\csc^2{\left(x\right)}\right)
\\&=&-2\csc{\left(x\right)}\frac{\mathrm{d}}{\mathrm{d}x}\csc{\left(x\right)}
\;\cdots\;u=\csc{\left(x\right)},f=-u^2,\frac{\mathrm{d}f}{\mathrm{d}x}=\frac{\mathrm{d}f}{\mathrm{d}u}\frac{\mathrm{d}u}{\mathrm{d}x}=-2u\frac{\mathrm{d}u}{\mathrm{d}x}
\\&=&-2\csc{\left(x\right)}\left(-\csc{\left(x\right)}\cot{\left(x\right)}\right)\;\cdots\;準備\;参照
\\&=&2\csc^2{\left(x\right)}\cot{\left(x\right)}
\;\cdots\;\frac{\mathrm{d}}{\mathrm{d}x}\csc^2{\left(x\right)}=-2\csc^2{\left(x\right)}\cot{\left(x\right)}でもある(後で使う)．
\\\;
\\\frac{\mathrm{d}^3}{\mathrm{d}x^3}\cot{\left(x\right)}
&=&\frac{\mathrm{d}}{\mathrm{d}x}\left(2\csc^2{\left(x\right)}\cot{\left(x\right)}\right)
\\&=&2\frac{\mathrm{d}}{\mathrm{d}x}\left(\csc^2{\left(x\right)}\cot{\left(x\right)}\right)
\\&=&2\left\{
\left(\frac{\mathrm{d}}{\mathrm{d}x}\csc^2{\left(x\right)}\right)\cot{\left(x\right)}
+\csc^2{\left(x\right)}\left(\frac{\mathrm{d}}{\mathrm{d}x}\cot{\left(x\right)}\right)
\right\}
\;\cdots\;(uv)^\prime=u^\prime v+u v^\prime
\\&=&2\left\{
\left(-2\csc^2{\left(x\right)}\cot{\left(x\right)}\right)\cot{\left(x\right)}
+\csc^2{\left(x\right)}\left(-\csc^2{\left(x\right)}\right)
\right\}
\;\cdots\;\frac{\mathrm{d}}{\mathrm{d}x}\csc^2{\left(x\right)}=-2\csc^2{\left(x\right)}\cot{\left(x\right)}
,\frac{\mathrm{d}}{\mathrm{d}x}\cot{\left(x\right)}=-\csc^2{\left(x\right)}
\\&=&-2\left(
2\csc^2{\left(x\right)}\cot^2{\left(x\right)}
+\csc^4{\left(x\right)}
\right)
\\\;
\\\frac{\mathrm{d}^4}{\mathrm{d}x^4}\cot{\left(x\right)}
&=&\frac{\mathrm{d}}{\mathrm{d}x}
\left(-2\left(
2\csc^2{\left(x\right)}\cot^2{\left(x\right)}
+\csc^4{\left(x\right)}
\right)\right)
\\&=&-2\left[
\frac{\mathrm{d}}{\mathrm{d}x}\left(
2\csc^2{\left(x\right)}\cot^2{\left(x\right)}
+\csc^4{\left(x\right)}
\right)\right]
\\&=&-2\left[
\frac{\mathrm{d}}{\mathrm{d}x}\left(
2\csc^2{\left(x\right)}\cot^2{\left(x\right)}
\right)
+\frac{\mathrm{d}}{\mathrm{d}x}\left(
\csc^4{\left(x\right)}
\right)
\right]
\;\cdots\;(u+v)^\prime=u^\prime+v^\prime
\\&=&-2\left[2\left\{
\left(\frac{\mathrm{d}}{\mathrm{d}x}\csc^2{\left(x\right)}\right) \cot^2{\left(x\right)}
+\csc^2{\left(x\right)}\left(\frac{\mathrm{d}}{\mathrm{d}x}\cot^2{\left(x\right)}\right)
\right\}
+
4\csc^3{\left(x\right)}
\left(
-\csc{\left(x\right)}\cot{\left(x\right)}
\right)
\right]
\;\cdots\;(uv)^\prime=u^\prime v+u v^\prime
,u=\csc{\left(x\right)},f=u^4,\frac{\mathrm{d}f}{\mathrm{d}x}=\frac{\mathrm{d}f}{\mathrm{d}u}\frac{\mathrm{d}u}{\mathrm{d}x}=4u^3\frac{\mathrm{d}u}{\mathrm{d}x}
,\frac{\mathrm{d}}{\mathrm{d}x}\csc{\left(x\right)}=-\csc{\left(x\right)}\cot{\left(x\right)}
\\&=&-2\left[2\left\{
(-2\csc^2{\left(x\right)}\cot{\left(x\right)}) \cot^2{\left(x\right)}
+\csc^2{\left(x\right)}(2\cot{\left(x\right)}\left(-\csc^2{\left(x\right)}\right))
\right\}
-4\csc^4{\left(x\right)}\cot{\left(x\right)}
\right]
\;\cdots\;\frac{\mathrm{d}}{\mathrm{d}x}\csc^2{\left(x\right)}=-2\csc^2{\left(x\right)}\cot{\left(x\right)}
,u=\cot{\left(x\right)},f=u^2,\frac{\mathrm{d}f}{\mathrm{d}x}=\frac{\mathrm{d}f}{\mathrm{d}u}\frac{\mathrm{d}u}{\mathrm{d}x}=2u\frac{\mathrm{d}u}{\mathrm{d}x}
,\frac{\mathrm{d}}{\mathrm{d}x}\cot{\left(x\right)}=-\csc^2{\left(x\right)}
\\&=&-2\left[2\left\{
-2\csc^2{\left(x\right)}\cot^3{\left(x\right)}
-2\csc^4{\left(x\right)}\cot{\left(x\right)}
\right\}
-4\csc^4{\left(x\right)}\cot{\left(x\right)}
\right]
\\&=&-2\left[
-4\csc^2{\left(x\right)}\cot^3{\left(x\right)}
-4\csc^4{\left(x\right)}\cot{\left(x\right)}
-4\csc^4{\left(x\right)}\cot{\left(x\right)}
\right]
\\&=&-2\left[
-4\csc^2{\left(x\right)}\cot^3{\left(x\right)}
-8\csc^4{\left(x\right)}\cot{\left(x\right)}
\right]
\\&=&8\csc^2{\left(x\right)}\cot{\left(x\right)}\left(\cot^2{\left(x\right)}+2\csc^2{\left(x\right)}\right)
\\\;
\\\frac{\mathrm{d}^5}{\mathrm{d}x^5}\cot{\left(x\right)}
&=&\frac{\mathrm{d}}{\mathrm{d}x}\left\{
8\csc^2{\left(x\right)}\cot{\left(x\right)}\left(\cot^2{\left(x\right)}+2\csc^2{\left(x\right)}\right)
\right\}
\\&=&4\frac{\mathrm{d}}{\mathrm{d}x} \left\{
2\csc^2{\left(x\right)}\cot{\left(x\right)}\left(\cot^2{\left(x\right)}+2\csc^2{\left(x\right)}\right)
\right\}
\\&=&4\left\{
\left(\frac{\mathrm{d}}{\mathrm{d}x}2\csc^2{\left(x\right)}\cot{\left(x\right)}\right)
\left(\cot^2{\left(x\right)}+2\csc^2{\left(x\right)}\right)
+
2\csc^2{\left(x\right)}\cot{\left(x\right)}
\left(\frac{\mathrm{d}}{\mathrm{d}x}\left(\cot^2{\left(x\right)}+2\csc^2{\left(x\right)}\right)\right)
\right\}
\\&=&4\left[
\left\{-2\left(
2\csc^2{\left(x\right)}\cot^2{\left(x\right)}
+\csc^4{\left(x\right)}
\right)\right\}
\left(\cot^2{\left(x\right)}+2\csc^2{\left(x\right)}\right)
+
2\csc^2{\left(x\right)}\cot{\left(x\right)}
\left(\frac{\mathrm{d}}{\mathrm{d}x}\cot^2{\left(x\right)}+2\frac{\mathrm{d}}{\mathrm{d}x}\csc^2{\left(x\right)}\right)
\right]
\\&=&4\left[
\left(
-4\csc^2{\left(x\right)}\cot^2{\left(x\right)}
-2\csc^4{\left(x\right)}
\right)
\left(\cot^2{\left(x\right)}+2\csc^2{\left(x\right)}\right)
+
2\csc^2{\left(x\right)}\cot{\left(x\right)}
\left\{
\left(-2\csc^2{\left(x\right)}\cot{\left(x\right)}\right)
+2\left(-2\csc^2{\left(x\right)}\cot{\left(x\right)}\right)
\right\}
\right]
\\&=&4\left\{
\left(
-4\csc^2{\left(x\right)}\cot^2{\left(x\right)}\left(\cot^2{\left(x\right)}+2\csc^2{\left(x\right)}\right)
-2\csc^4{\left(x\right)}\left(\cot^2{\left(x\right)}+2\csc^2{\left(x\right)}\right)
\right)
+
2\csc^2{\left(x\right)}\cot{\left(x\right)}\cdot-2\csc^2{\left(x\right)}\cot{\left(x\right)}
+2\csc^2{\left(x\right)}\cot{\left(x\right)}\cdot2\left(-2\csc^2{\left(x\right)}\cot{\left(x\right)}\right)
\right\}
\\&=&4\left(
-4\csc^2{\left(x\right)}\cot^4{\left(x\right)}-8\csc^4{\left(x\right)}\cot^2{\left(x\right)}
-2\csc^4{\left(x\right)}\cot^2{\left(x\right)}-4\csc^6{\left(x\right)}
-4\csc^4{\left(x\right)}\cot^2{\left(x\right)}
-8\csc^4{\left(x\right)}\cot^2{\left(x\right)}
\right)
\\&=&4\left(
-4\csc^2{\left(x\right)}\cot^4{\left(x\right)}
-4\csc^6{\left(x\right)}
-22\csc^4{\left(x\right)}\cot^2{\left(x\right)}
\right)
\\&=&-8\left(
2\csc^6{\left(x\right)}
+2\csc^2{\left(x\right)}\cot^4{\left(x\right)}
+11\csc^4{\left(x\right)}\cot^2{\left(x\right)}
\right)
\end{eqnarray}$$
\(x=\frac{\pi}{2}\)でのテーラー展開を求めておく
$$\begin{eqnarray}
\cot{\left(x\right)}&=&
\frac{1}{0!}\left[\left.\frac{\mathrm{d}^0}{\mathrm{d}x^0}\cot{\left(x\right)}\right|_{x=\frac{\pi}{2}}\right]\left(x-\frac{\pi}{2}\right)^0
\\&&+\frac{1}{1!}\left[\left.\frac{\mathrm{d}^1}{\mathrm{d}x^1}\cot{\left(x\right)}\right|_{x=\frac{\pi}{2}}\right]\left(x-\frac{\pi}{2}\right)^1
\\&&+\frac{1}{2!}\left[\left.\frac{\mathrm{d}^2}{\mathrm{d}x^2}\cot{\left(x\right)}\right|_{x=\frac{\pi}{2}}\right]\left(x-\frac{\pi}{2}\right)^2
\\&&+\frac{1}{3!}\left[\left.\frac{\mathrm{d}^3}{\mathrm{d}x^3}\cot{\left(x\right)}\right|_{x=\frac{\pi}{2}}\right]\left(x-\frac{\pi}{2}\right)^3
\\&&+\frac{1}{4!}\left[\left.\frac{\mathrm{d}^4}{\mathrm{d}x^4}\cot{\left(x\right)}\right|_{x=\frac{\pi}{2}}\right]\left(x-\frac{\pi}{2}\right)^4
\\&&+\frac{1}{5!}\left[\left.\frac{\mathrm{d}^5}{\mathrm{d}x^5}\cot{\left(x\right)}\right|_{x=\frac{\pi}{2}}\right]\left(x-\frac{\pi}{2}\right)^5
\\&&+\cdots
\\&=&
\frac{1}{1} \left[\cot{\left(\frac{\pi}{2}\right)}\right]\left(x-\frac{\pi}{2}\right)^0
\\&&+\frac{1}{1}\left[-\csc^2{\left(\frac{\pi}{2}\right)}\right]\left(x-\frac{\pi}{2}\right)^1
\\&&+\frac{1}{2}\left[2\cot{\left(\frac{\pi}{2}\right)}\csc^2{\left(\frac{\pi}{2}\right)}\right]\left(x-\frac{\pi}{2}\right)^2
\\&&+\frac{1}{6}\left[-2\left(2\csc^2{\left(\frac{\pi}{2}\right)}\cot^2{\left(\frac{\pi}{2}\right)}+\csc^4{\left(\frac{\pi}{2}\right)}\right)\right]\left(x-\frac{\pi}{2}\right)^3
\\&&+\frac{1}{24}\left[8\csc^2{\left(\frac{\pi}{2}\right)}\cot{\left(\frac{\pi}{2}\right)}\left(\cot^2{\left(\frac{\pi}{2}\right)}+2\csc^2{\left(\frac{\pi}{2}\right)}\right)\right]\left(x-\frac{\pi}{2}\right)^4
\\&&+\frac{1}{120}\left[-8\left(
2\csc^6{\left(\frac{\pi}{2}\right)}
+2\csc^2{\left(\frac{\pi}{2}\right)}\cot^4{\left(\frac{\pi}{2}\right)}
+11\csc^4{\left(\frac{\pi}{2}\right)}\cot^2{\left(\frac{\pi}{2}\right)}
\right)\right]\left(x-\frac{\pi}{2}\right)^5
\\&&+\cdots
\\&=&
\left[0\right]\cdot 1
\\&&+\left[-1\cdot1^2\right]\left(x-\frac{\pi}{2}\right)
\\&&+\frac{1}{2}\left[2\cdot 0 \cdot 1^2\right]\left(x-\frac{\pi}{2}\right)^2
\\&&+\frac{1}{6}\left[-2\left(2 \cdot 1^2 \cdot 0^2+ 1^4\right)\right]\left(x-\frac{\pi}{2}\right)^3
\\&&+\frac{1}{24}\left[8\cdot1^2 \cdot0\left(0^2+2\cdot1^2\right)\right]\left(x-\frac{\pi}{2}\right)^4
\\&&+\frac{1}{120}\left[-8\left(2\cdot 1^6+2\cdot 1^2\cdot0^4
+11\cdot1^4\cdot0^2\right)\right]\left(x-\frac{\pi}{2}\right)^5
\\&&+\cdots
\\&=&
-\left(x-\frac{\pi}{2}\right)-\frac{2}{6}\left(x-\frac{\pi}{2}\right)^3-\frac{16}{120}\left(x-\frac{\pi}{2}\right)^5+\cdots
\\&=&
-\left(x-\frac{\pi}{2}\right)-\frac{1}{3}\left(x-\frac{\pi}{2}\right)^3-\frac{2}{15}\left(x-\frac{\pi}{2}\right)^5+\cdots
\end{eqnarray}$$
\(\lim_{x\rightarrow \frac{\pi}{2}} \cot{\left(x\right)}\)を求める
$$\begin{eqnarray}
\\\lim_{x\rightarrow \frac{\pi}{2}} \cot{\left(x\right)}&=&
\lim_{x\rightarrow \frac{\pi}{2}} \left[
-\left(x-\frac{\pi}{2}\right)-\frac{1}{3}\left(x-\frac{\pi}{2}\right)^3-\frac{2}{15}\left(x-\frac{\pi}{2}\right)^5+\cdots
\right]
\\&=&0
\end{eqnarray}$$
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