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人気のある微分積分 >

limit as x approaches 0 of 2x^2*cot^2(x)

解

x \to 0 lim (2 x^{2} \cdot cot^{2} (x))

解

2

解答ステップ

x \to 0 lim (2 x^{2} cot^{2} (x))

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

= 2 \cdot x \to 0 lim (x^{2} cot^{2} (x))

指数の規則を適用する:

a^{n} \cdot b^{n} = (a \cdot b)^{n}

x^{2} cot^{2} (x) = (x cot (x))^{2}

= 2 \cdot x \to 0 lim ((x cot (x))^{2})

x \to a lim [f (x)]^{b} = [x \to a lim f (x)]^{b}

不定形式の場合を除く

= 2 (x \to 0 lim (x cot (x)))^{2}

ロピタル向けに書き換える

= 2 (x \to 0 lim (\frac{x}{\frac{1}{c o t ( x )}}))^{2}

ロピタルの定理を適用する

= 2 (x \to 0 lim (\frac{1}{sec ^{2} ( x )}))^{2}

値を挿入する

x = 0

= 2 (\frac{1}{sec ^{2} ( 0 )})^{2}

簡素化

2 (\frac{1}{sec ^{2} ( 0 )})^{2} : 2

= 2

グラフ

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limit as x approaches 0 of 3x*cot(3x)

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