\lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f'(x)}{g'(x)}

\lim_{x\to t}\,f(x) = f(t)

f'(x) = \frac{x\cos x -\sin x}{x^2}

\begin{align}
f'(0) 
&= \lim_{h\to 0} \frac{f(h) - f(0)}{h} \\
&= \lim_{h\to 0} \frac{\frac{\sin h}{h} - 1}{h} \\
&= \lim_{h\to 0} \frac{\sin h - h}{h^2} \\
&= \lim_{h\to 0} \frac{\cos h}{2h} &&\text{（ロピタルの定理）} \\
&= \lim_{h\to 0} \frac{-\sin h}{2} &&\text{（ロピタルの定理）} \\
&= 0
\end{align}

f'(x) = \begin{cases}
\frac{x\cos x -\sin x}{x^2} && (x\not=0) \\
0 && (x=0)
\end{cases}

\begin{align}
\lim_{x\to 0}\frac{x\cos x -\sin x}{x^2} 
&= \lim_{x\to 0}\frac{\cos x -x\sin x -\cos x}{2x}  &&\text{（ロピタルの定理）} \\
&= \lim_{x\to 0} \frac{-x\sin x}{2x}   \\
&= \lim_{x\to 0} \frac{-\sin x}{2}   \\
&= 0 \\
&= f'(0).
\end{align}

\begin{align}
a_n &= f'(n\pi) \\
&= \frac{n\pi\cos (n\pi) -\sin (n\pi)}{(n\pi)^2} \\
&= \frac{n\pi\cos (n\pi)}{(n\pi)^2} && \text{($\sin \pi, \sin 2\pi, \dots = 0$)} \\
&= \frac{\cos (n\pi)}{n\pi}  
\end{align}

\sum_{n=1}^{\infty} a_n 
= \frac{1}{\pi} \left( -1 + \frac{1}{2}  - \frac{1}{3} + \frac{1}{4} - \cdots  \right)

\begin{align}
\log(x) &= 0 + (x-1) - \frac{1}{2!} (x-1)^2 + \frac{2}{3!}(x-1)^3 - \frac{3\cdot 2}{4!}(x-1)^4 + \cdots \\
&= (x-1) - \frac{1}{2} (x-1)^2 + \frac{1}{3}(x-1)^3 - \frac{1}{4}(x-1)^4 + \cdots
\end{align}

\begin{align}
\sum_{n=1}^{\infty} a_n 
&= \frac{1}{\pi} \left( -1 + \frac{1}{2}  - \frac{1}{3} + \frac{1}{4} - \cdots  \right) \\
&= \frac{-\log(2)}{\pi}.
\end{align}