limit as x approaches infinity of sqrt(x^2+7x+2)-x

解

x \to \infty lim (x^{2} + 7 x + 2 - x)

\frac{7}{2}

十進法表記

3.5

解答ステップ

x \to \infty lim (x^{2} + 7 x + 2 - x)

x^{2} + 7 x + 2 - x

の共役で乗じる

\frac{7 x + 2}{x ^{2} + 7 x + 2 + x}

= x \to \infty lim (\frac{7 x + 2}{x ^{2} + 7 x + 2 + x})

最高の分母べき乗で割る:

\frac{7 + \frac{2}{x}}{1 + \frac{7}{x} + \frac{2}{x ^{2}} + 1}

= x \to \infty lim \frac{7 + \frac{2}{x}}{1 + \frac{7}{x} + \frac{2}{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

= \frac{lim _{x \to \infty} ( 7 + \frac{2}{x} )}{lim _{x \to \infty} ( 1 + \frac{7}{x} + \frac{2}{x ^{2}} + 1 )}

x \to \infty lim (7 + \frac{2}{x}) = 7

x \to \infty lim (1 + \frac{7}{x} + \frac{2}{x ^{2}} + 1) = 2

= \frac{7}{2}