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人気のある微分積分 >

limit as x approaches 0 of (e^{x^2}-cos(x))/(sin^2(x)+3sin^5(x))

解

x \to 0 lim (\frac{e ^{x^{2}} - cos ( x )}{sin ^{2} ( x ) + 3 sin ^{5} ( x )})

解

\frac{3}{2}

+1

十進法表記

1.5

解答ステップ

x \to 0 lim (\frac{e ^{x^{2}} - cos ( x )}{sin ^{2} ( x ) + 3 sin ^{5} ( x )})

ロピタルの定理を適用する

= x \to 0 lim (\frac{2 e ^{x^{2}} x + sin ( x )}{sin ( 2 x ) + 15 sin ^{4} ( x ) cos ( x )})

ロピタルの定理を適用する

= x \to 0 lim \frac{2 ( 2 e ^{x^{2}} x ^{2} + e ^{x^{2}} ) + cos ( x )}{cos ( 2 x ) \cdot 2 + 15 ( 4 sin ^{3} ( x ) cos ^{2} ( x ) - sin ^{5} ( x ) )}

値を挿入する

x = 0

= \frac{2 ( 2 e ^{0^{2}} \cdot 0 ^{2} + e ^{0^{2}} ) + cos ( 0 )}{cos ( 2 \cdot 0 ) \cdot 2 + 15 ( 4 sin ^{3} ( 0 ) cos ^{2} ( 0 ) - sin ^{5} ( 0 ) )}

簡素化

\frac{2 ( 2 e ^{0^{2}} \cdot 0 ^{2} + e ^{0^{2}} ) + cos ( 0 )}{cos ( 2 \cdot 0 ) \cdot 2 + 15 ( 4 sin ^{3} ( 0 ) cos ^{2} ( 0 ) - sin ^{5} ( 0 ) )} : \frac{3}{2}

= \frac{3}{2}

グラフ

人気の例

limit as x approaches-2 of sqrt(3x^2+4)

x \to - 2 lim (3 x^{2} + 4)

limit as x approaches-1+of 2x+1

x \to - 1 + lim (2 x + 1)

limit as x approaches 0+of (1+5x)^{2/x}

x \to 0 + lim ((1 + 5 x)^{\frac{2}{x}})

limit as x approaches 1 of (3x)/(x-1)

x \to 1 lim (\frac{3 x}{x - 1})

limit as x approaches 1/5 of 2+(5x-1)

x \to \frac{1}{5} lim (2 + (5 x - 1))