limit as x approaches infinity of (3x^6+3x^2+2)/(7x^6+x-1)

解

x \to \infty lim (\frac{3 x ^{6} + 3 x ^{2} + 2}{7 x ^{6} + x - 1})

\frac{3}{7}

十進法表記

0.42857 \dots

解答ステップ

x \to \infty lim (\frac{3 x ^{6} + 3 x ^{2} + 2}{7 x ^{6} + x - 1})

最高の分母べき乗で割る:

\frac{3 + \frac{3}{x ^{4}} + \frac{2}{x ^{6}}}{7 + \frac{1}{x ^{5}} - \frac{1}{x ^{6}}}

= x \to \infty lim (\frac{3 + \frac{3}{x ^{4}} + \frac{2}{x ^{6}}}{7 + \frac{1}{x ^{5}} - \frac{1}{x ^{6}}})

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} ( 3 + \frac{3}{x ^{4}} + \frac{2}{x ^{6}} )}{lim _{x \to \infty} ( 7 + \frac{1}{x ^{5}} - \frac{1}{x ^{6}} )}

x \to \infty lim (3 + \frac{3}{x ^{4}} + \frac{2}{x ^{6}}) = 3

x \to \infty lim (7 + \frac{1}{x ^{5}} - \frac{1}{x ^{6}}) = 7

= \frac{3}{7}