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人気のある微分積分 >

limit as x approaches infinity of (((log_{2}(x))^2))/(sqrt(x))

解

x \to \infty lim \frac{( ( lo g _{2} ( x ) ) ^{2} )}{x}

解

0

解答ステップ

x \to \infty lim (\frac{( lo g _{2} ( x ) ) ^{2}}{x})

改良

= x \to \infty lim (\frac{lo g _{2} ( x ) ^{2}}{x})

ロピタルの定理を適用する

= x \to \infty lim \frac{\frac{2 l o g _{2} ( x )}{l n ( 2 ) x}}{\frac{1}{2 x}}

簡素化

\frac{\frac{2 l o g _{2} ( x )}{l n ( 2 ) x}}{\frac{1}{2 x}} : \frac{4 lo g _{2} ( x )}{ln ( 2 ) x}

= x \to \infty lim (\frac{4 lo g _{2} ( x )}{ln ( 2 ) x})

ロピタルの定理を適用する

= x \to \infty lim \frac{\frac{4}{l n ( 2 ) x}}{\frac{l n ( 2 )}{2 x}}

簡素化

\frac{\frac{4}{l n ( 2 ) x}}{\frac{l n ( 2 )}{2 x}} : \frac{8}{ln ^{2} ( 2 ) x}

= x \to \infty lim (\frac{8}{ln ^{2} ( 2 ) x})

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

= \frac{8}{ln ^{2} ( 2 )} \cdot x \to \infty lim (\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{8}{ln ^{2} ( 2 )} \cdot \frac{lim _{x \to \infty} ( 1 )}{lim _{x \to \infty} ( x )}

x \to \infty lim (1) = 1

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

= \frac{8}{ln ^{2} ( 2 )} \cdot \frac{1}{\infty}

簡素化

\frac{8}{ln ^{2} ( 2 )} \cdot \frac{1}{\infty} : 0

= 0

グラフ

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