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人気のある微分積分 >

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

解

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

解

2

解答ステップ

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

x^{2} + 4 - x

の共役で乗じる

\frac{4}{x ^{2} + 4 + x}

= x \to \infty lim (x \frac{4}{x ^{2} + 4 + x})

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

= 4 \cdot x \to \infty lim (x \frac{1}{x ^{2} + 4 + x})

簡素化

x \frac{1}{x ^{2} + 4 + x} : \frac{x}{x ^{2} + 4 + x}

= 4 \cdot x \to \infty lim (\frac{x}{x ^{2} + 4 + x})

最高の分母べき乗で割る:

\frac{1}{1 + \frac{4}{x ^{2}} + 1}

= 4 \cdot x \to \infty lim \frac{1}{1 + \frac{4}{x ^{2}} + 1}

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

= 4 \cdot \frac{lim _{x \to \infty} ( 1 )}{lim _{x \to \infty} ( 1 + \frac{4}{x ^{2}} + 1 )}

x \to \infty lim (1) = 1

x \to \infty lim (1 + \frac{4}{x ^{2}} + 1) = 2

= 4 \cdot \frac{1}{2}

簡素化

4 \cdot \frac{1}{2} : 2

= 2

グラフ

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