lim x→π/2 cot(x) を求める
\(\lim_{x\rightarrow \frac{\pi}{2}} \cot{\left(x\right)}\)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
$$\begin{eqnarray} \\\lim_{x\rightarrow \frac{\pi}{2}} \cot{\left(x\right)}&=& \lim_{x\rightarrow \frac{\pi}{2}} \left[ -\left(x-\frac{\pi}{2}\right)-\frac{1}{3}\left(x-\frac{\pi}{2}\right)^3-\frac{2}{15}\left(x-\frac{\pi}{2}\right)^5+\cdots \right] \\&=&0 \end{eqnarray}$$(x^n)/(e^x)の極限
(x^n)/(e^x)の極限
以下を証明する $$ \begin{eqnarray} \lim_{x\rightarrow\infty}\frac{x^n}{e^x}&=&0 \end{eqnarray} $$ まず\(e^x\)のマクローリン展開から\(e^x\)より小さい値を考える． $$ \begin{eqnarray} \\e^x &=&\href{https://shikitenkai.blogspot.com/2019/07/blog-post.html}{\sum_{k=0}^{\infty}\frac{x^k}{k!}} \\&=&\sum_{k=0}^{\infty}f_k(x) \;\cdots\;f_k(x)=\frac{x^k}{k!} \\&\gt&\frac{x^{n+1}}{(n+1)!}\;\cdots\;いくつものf_kを足し合わせたe^xの方が特定のf_k(例えばk=n+1)だけより大きい \end{eqnarray} $$ 上記値の逆数をとる． $$ \begin{eqnarray} \\\frac{1}{e^x}&\lt&\frac{(n+1)!}{x^{n+1}} \;\cdots\;逆数なので大小関係が反対になる \end{eqnarray} $$ 両辺にx^nを掛け，証明したい式の形にする． $$ \begin{eqnarray} \\x^n\frac{1}{e^x}&\lt&x^n\frac{(n+1)!}{x^{n+1}} \;\cdots\;両辺にx^nを掛ける \\&\lt&\frac{(n+1)!}{x} \;\cdots\;\frac{x^n}{x^{n+1}}=\frac{1}{x} \end{eqnarray} $$ 両辺の極限をとることで\(\frac{x^n}{e^x}\)の極限の値が0より小さいことがわかる． $$ \begin{eqnarray} \\\lim_{x\rightarrow\infty}\frac{x^n}{e^x}&\lt&\lim_{x\rightarrow\infty}\frac{(n+1)!}{x}&=&0 \end{eqnarray} $$ また，\(0\leq x\)において\(0\leq x^n\)及び\(0\lt e^x\)となるので\(\frac{x^n}{e^x}\)が0以上であることがわかる．以上から次の関係（不等式）を得る． $$ \begin{eqnarray} 0&\leq& \lim_{x\rightarrow\infty}\frac{x^n}{e^x} &\lt&\lim_{x\rightarrow\infty}\frac{(n+1)!}{x} &=&0 \end{eqnarray} $$ これより“はさみうちの原理”から極限の値が0であることが証明された． $$ \begin{eqnarray} \\\lim_{x\rightarrow\infty}\frac{x^n}{e^x}&=&0 \end{eqnarray} $$
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