<s>[INST] Find any rational approximation of $a = \frac{\sin(\sqrt{2})}{\sqrt{2}}$ I need to find any ratinal approximation of $a = \frac{\sin(\sqrt{2})}{\sqrt{2}} $ with accuracy $d = \frac{1}{500}$.
I generally know how to do these type of tasks (using Taylor expansion), but I got problem with this one. 
I defined function $f(x) = \begin{cases} 1 \ \mbox{dla} \ x=0 \\ \frac{\sin x}{x} \ \mbox{dla} \ x \neq 0 \end{cases}$, to handle problem with dividng by zero in $f(0)$.
I calculate first and second derivative:
$f'(x) = \frac{x^2 \cdot \cos x - \sin x}{x^2}$
$f''(x) = \frac{x^3 \cdot \sin x - 2x^2 \cdot \cos x + \sin x }{x^4}$
But well, what now? I have no idea how to use the Lagrange remainder theorem here.
 [/INST] $$\frac{1}{x}\sin x=\frac{1}{x}(x-\frac{x^3}{3!}+\frac{x^5}{5!}...)=1-\frac{x^2}{3!}+\frac{x^4}{5!}...$$
hence
$$=$$
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