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人気のある微分積分 >

limit as x approaches infinity of x(arctan(x)-pi/2)

解

x \to \infty lim (x (arctan (x) - \frac{π}{2}))

解

- 1

解答ステップ

x \to \infty lim (x (arctan (x) - \frac{π}{2}))

ロピタル向けに書き換える

= x \to \infty lim (\frac{arctan ( x ) - \frac{π}{2}}{\frac{1}{x}})

ロピタルの定理を適用する

= x \to \infty lim (\frac{\frac{1}{x ^{2} + 1}}{- \frac{1}{x ^{2}}})

簡素化

\frac{\frac{1}{x ^{2} + 1}}{- \frac{1}{x ^{2}}} : - \frac{x ^{2}}{x ^{2} + 1}

= x \to \infty lim (- \frac{x ^{2}}{x ^{2} + 1})

x \to a lim [c \cdot f (x)] = c \cdot x \to a lim f (x)

= - x \to \infty lim (\frac{x ^{2}}{x ^{2} + 1})

最高の分母べき乗で割る:

\frac{1}{1 + \frac{1}{x ^{2}}}

= - x \to \infty lim (\frac{1}{1 + \frac{1}{x ^{2}}})

x \to a lim [\frac{f ( x )}{g ( x )}] = \frac{lim _{x \to a} f ( x )}{lim _{x \to a} g ( x )}, x \to a lim g (x) \neq = 0

不定形式の場合を除く

= - \frac{lim _{x \to \infty} ( 1 )}{lim _{x \to \infty} ( 1 + \frac{1}{x ^{2}} )}

x \to \infty lim (1) = 1

x \to \infty lim (1 + \frac{1}{x ^{2}}) = 1

= - \frac{1}{1}

簡素化

= - 1

グラフ

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