Wyznaczyć wartość granicy (egzaminacyjne)

[Dzikster]

\(\displaystyle{ \lim_{ x\to \infty } (xarctg(1+ \frac{1}{x})- \frac{Pi}{4}x)}\)

[meninio]

\(\displaystyle{ \lim_{ x \to \infty} \frac{\arctan \left(1+\frac{1}{x} \right) }{\frac{1}{x}}-\frac{\pi}{\frac{4}{x}} = \lim_{ x \to \infty} \frac{4\arctan \left(1+\frac{1}{x} \right) -\pi }{\frac{4}{x}} = \left[ \frac{0}{0} \right] \stackrel{[H]}{=} \lim_{ x \to \infty} \frac{4 \cdot \frac{-1}{x^2} \cdot \frac{1}{1+ \left(1+\frac{1}{x} \right) ^2}}{-\frac{4}{x^2}} = \\ \\ = \lim_{ x \to \infty} \frac{1}{1+ \left(1+\frac{1}{x} \right) ^2} = \frac{1}{2}}\)