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人気のある微分積分 >

limit as x approaches 0 of [ 1/(x^2)-1/(ln(1+x^2))]

解

x \to 0 lim ([\frac{1}{x ^{2}} - \frac{1}{ln ( 1 + x ^{2} )}])

解

- \frac{1}{2}

+1

十進法表記

- 0.5

解答ステップ

x \to 0 lim (\frac{1}{x ^{2}} - \frac{1}{ln ( 1 + x ^{2} )})

簡素化

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

= x \to 0 lim (\frac{ln ( x ^{2} + 1 ) - x ^{2}}{x ^{2} ln ( x ^{2} + 1 )})

ロピタルの定理を適用する

= x \to 0 lim (\frac{- \frac{2 x ^{3}}{x ^{2} + 1}}{2 x ln ( x ^{2} + 1 ) + \frac{2 x ^{3}}{x ^{2} + 1}})

簡素化

\frac{- \frac{2 x ^{3}}{x ^{2} + 1}}{2 x ln ( x ^{2} + 1 ) + \frac{2 x ^{3}}{x ^{2} + 1}} : - \frac{x ^{2}}{ln ( x ^{2} + 1 ) ( x ^{2} + 1 ) + x ^{2}}

= x \to 0 lim (- \frac{x ^{2}}{ln ( x ^{2} + 1 ) ( x ^{2} + 1 ) + x ^{2}})

ロピタルの定理を適用する

= x \to 0 lim (\frac{- 2 x}{4 x + 2 x ln ( x ^{2} + 1 )})

簡素化

\frac{- 2 x}{4 x + 2 x ln ( x ^{2} + 1 )} : - \frac{1}{2 + ln ( x ^{2} + 1 )}

= x \to 0 lim (- \frac{1}{2 + ln ( x ^{2} + 1 )})

値を挿入する

x = 0

= - \frac{1}{2 + ln ( 0 ^{2} + 1 )}

簡素化

- \frac{1}{2 + ln ( 0 ^{2} + 1 )} : - \frac{1}{2}

= - \frac{1}{2}

グラフ

人気の例

limit as x approaches 4-of (ax+b)/(x+c)

x \to 4 - lim (\frac{a x + b}{x + c})

limit as x approaches infinity of sqrt(x^2+3x+19)-x

x \to \infty lim (x^{2} + 3 x + 19 - x)

limit as x approaches 0 of x^3+x^2+3

x \to 0 lim (x^{3} + x^{2} + 3)

limit as x approaches pi/6 of sec^2(x)

x \to \frac{π}{6} lim (sec^{2} (x))

limit as x approaches infinity of x*10

x \to \infty lim (x \cdot 10)