limit as x approaches infinity of (4x^3+8)/(4x^3+sqrt(64x^6+3))

解

x \to \infty lim (\frac{4 x ^{3} + 8}{4 x ^{3} + 64 x ^{6} + 3})

\frac{1}{3}

十進法表記

0.33333 \dots

解答ステップ

x \to \infty lim (\frac{4 x ^{3} + 8}{4 x ^{3} + 64 x ^{6} + 3})

最高の分母べき乗で割る:

\frac{4 + \frac{8}{x ^{3}}}{4 + 64 + \frac{3}{x ^{6}}}

= x \to \infty lim \frac{4 + \frac{8}{x ^{3}}}{4 + 64 + \frac{3}{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} ( 4 + \frac{8}{x ^{3}} )}{lim _{x \to \infty} ( 4 + 64 + \frac{3}{x ^{6}} )}

x \to \infty lim (4 + \frac{8}{x ^{3}}) = 4

x \to \infty lim (4 + 64 + \frac{3}{x ^{6}}) = 12

= \frac{4}{12}

共通因数を約分する:

4

= \frac{1}{3}