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人気のある微分積分 >

limit as x approaches-infinity of xlog_{e}(1-1/x)

解

x \to - \infty lim (x lo g_{e} (1 - \frac{1}{x}))

解

- 1

解答ステップ

x \to - \infty lim (x lo g_{e} (1 - \frac{1}{x}))

簡素化

lo g_{e} (1 - \frac{1}{x}) : ln (- \frac{1}{x} + 1)

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

ロピタル向けに書き換える

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

ロピタルの定理を適用する

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

簡素化

\frac{\frac{1}{x ( x - 1 )}}{- \frac{1}{x ^{2}}} : - \frac{x}{x - 1}

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

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

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

最高の分母べき乗で割る:

\frac{1}{1 - \frac{1}{x}}

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

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} )}

x \to - \infty lim (1) = 1

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

= - \frac{1}{1}

簡素化

= - 1

グラフ

人気の例

limit as x approaches 0 of 2-cot(x)

x \to 0 lim (2 - cot (x))

limit as x approaches infinity of 1-3x

x \to \infty lim (1 - 3 x)

limit as x approaches-pi of (x-pi)csc(x)

x \to - π lim ((x - π) csc (x))

limit as x approaches 5 of 3x-12

x \to 5 lim (3 x - 12)

limit as x approaches 1 of ln(1+x)

x \to 1 lim (ln (1 + x))