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人気のある微分積分 >

limit as x approaches 0 of (tan(4x)cos(6x)-tan(4x))/(24x^3)

解

x \to 0 lim (\frac{tan ( 4 x ) cos ( 6 x ) - tan ( 4 x )}{24 x ^{3}})

解

- 3

解答ステップ

x \to 0 lim (\frac{tan ( 4 x ) cos ( 6 x ) - tan ( 4 x )}{24 x ^{3}})

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

= \frac{1}{24} \cdot x \to 0 lim (\frac{tan ( 4 x ) cos ( 6 x ) - tan ( 4 x )}{x ^{3}})

ロピタルの定理を適用する

= \frac{1}{24} \cdot x \to 0 lim (\frac{4 sec ^{2} ( 4 x ) cos ( 6 x ) - 6 sin ( 6 x ) tan ( 4 x ) - sec ^{2} ( 4 x ) \cdot 4}{3 x ^{2}})

ロピタルの定理を適用する

= \frac{1}{24} \cdot x \to 0 lim (\frac{32 sec ^{2} ( 4 x ) tan ( 4 x ) cos ( 6 x ) - 48 sec ^{2} ( 4 x ) sin ( 6 x ) - 36 cos ( 6 x ) tan ( 4 x ) - 32 sec ^{2} ( 4 x ) tan ( 4 x )}{6 x})

ロピタルの定理を適用する

= \frac{1}{24} \cdot x \to 0 lim (\frac{256 sec ^{2} ( 4 x ) tan ^{2} ( 4 x ) cos ( 6 x ) + 128 sec ^{4} ( 4 x ) cos ( 6 x ) - 576 sec ^{2} ( 4 x ) tan ( 4 x ) sin ( 6 x ) - 432 sec ^{2} ( 4 x ) cos ( 6 x ) + 216 sin ( 6 x ) tan ( 4 x ) - 256 sec ^{2} ( 4 x ) tan ^{2} ( 4 x ) - 128 sec ^{4} ( 4 x )}{6})

値を挿入する

x = 0

= \frac{1}{24} \cdot \frac{256 sec ^{2} ( 4 \cdot 0 ) tan ^{2} ( 4 \cdot 0 ) cos ( 6 \cdot 0 ) + 128 sec ^{4} ( 4 \cdot 0 ) cos ( 6 \cdot 0 ) - 576 sec ^{2} ( 4 \cdot 0 ) tan ( 4 \cdot 0 ) sin ( 6 \cdot 0 ) - 432 sec ^{2} ( 4 \cdot 0 ) cos ( 6 \cdot 0 ) + 216 sin ( 6 \cdot 0 ) tan ( 4 \cdot 0 ) - 256 sec ^{2} ( 4 \cdot 0 ) tan ^{2} ( 4 \cdot 0 ) - 128 sec ^{4} ( 4 \cdot 0 )}{6}

簡素化

\frac{1}{24} \cdot \frac{256 sec ^{2} ( 4 \cdot 0 ) tan ^{2} ( 4 \cdot 0 ) cos ( 6 \cdot 0 ) + 128 sec ^{4} ( 4 \cdot 0 ) cos ( 6 \cdot 0 ) - 576 sec ^{2} ( 4 \cdot 0 ) tan ( 4 \cdot 0 ) sin ( 6 \cdot 0 ) - 432 sec ^{2} ( 4 \cdot 0 ) cos ( 6 \cdot 0 ) + 216 sin ( 6 \cdot 0 ) tan ( 4 \cdot 0 ) - 256 sec ^{2} ( 4 \cdot 0 ) tan ^{2} ( 4 \cdot 0 ) - 128 sec ^{4} ( 4 \cdot 0 )}{6} : - 3

= - 3

グラフ

人気の例

limit as x approaches+1 of (x-1)/(x^6-1)

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