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人気のある微分積分 >

limit as x approaches infinity of ln(x)sin(1/x)

解

x \to \infty lim (ln (x) sin (\frac{1}{x}))

解

0

解答ステップ

x \to \infty lim (ln (x) sin (\frac{1}{x}))

ロピタル向けに書き換える

= x \to \infty lim (\frac{sin ( \frac{1}{x} )}{\frac{1}{l n ( x )}})

ロピタルの定理を適用する

= x \to \infty lim \frac{- \frac{c o s ( \frac{1}{x} )}{x ^{2}}}{- \frac{1}{x l n ^{2} ( x )}}

簡素化

\frac{- \frac{c o s ( \frac{1}{x} )}{x ^{2}}}{- \frac{1}{x l n ^{2} ( x )}} : \frac{ln ^{2} ( x ) cos ( \frac{1}{x} )}{x}

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

ロピタルの定理を適用する

= x \to \infty lim \frac{\frac{2 l n ( x ) c o s ( \frac{1}{x} )}{x} + \frac{l n ^{2} ( x ) s i n ( \frac{1}{x} )}{x ^{2}}}{1}

簡素化

\frac{\frac{2 l n ( x ) c o s ( \frac{1}{x} )}{x} + \frac{l n ^{2} ( x ) s i n ( \frac{1}{x} )}{x ^{2}}}{1} : \frac{2 x ln ( x ) cos ( \frac{1}{x} ) + ln ^{2} ( x ) sin ( \frac{1}{x} )}{x ^{2}}

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

ロピタルの定理を適用する

= x \to \infty lim \frac{2 ln ( x ) cos ( \frac{1}{x} ) + 2 cos ( \frac{1}{x} ) + \frac{4 s i n ( \frac{1}{x} ) l n ( x )}{x} - \frac{l n ^{2} ( x ) c o s ( \frac{1}{x} )}{x ^{2}}}{2 x}

簡素化

\frac{2 ln ( x ) cos ( \frac{1}{x} ) + 2 cos ( \frac{1}{x} ) + \frac{4 s i n ( \frac{1}{x} ) l n ( x )}{x} - \frac{l n ^{2} ( x ) c o s ( \frac{1}{x} )}{x ^{2}}}{2 x} : \frac{2 x ^{2} ln ( x ) cos ( \frac{1}{x} ) + 2 x ^{2} cos ( \frac{1}{x} ) + 4 x sin ( \frac{1}{x} ) ln ( x ) - ln ^{2} ( x ) cos ( \frac{1}{x} )}{2 x ^{3}}

= x \to \infty lim (\frac{2 x ^{2} ln ( x ) cos ( \frac{1}{x} ) + 2 x ^{2} cos ( \frac{1}{x} ) + 4 x sin ( \frac{1}{x} ) ln ( x ) - ln ^{2} ( x ) cos ( \frac{1}{x} )}{2 x ^{3}})

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

= \frac{1}{2} \cdot x \to \infty lim (\frac{2 x ^{2} ln ( x ) cos ( \frac{1}{x} ) + 2 x ^{2} cos ( \frac{1}{x} ) + 4 x sin ( \frac{1}{x} ) ln ( x ) - ln ^{2} ( x ) cos ( \frac{1}{x} )}{x ^{3}})

挟み撃ちの原理を適用する:

0

= \frac{1}{2} \cdot 0

簡素化

= 0

グラフ

人気の例

limit as x approaches 0+of cos^{1/x}(3x)

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