limit as x approaches infinity of (4x^3-6x^2)/(7x^{7/2)-5x-3}

解

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

0

解答ステップ

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

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

で割る

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

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

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

x \to \infty lim (\frac{4}{x ^{\frac{1}{2}}} - \frac{6}{x ^{\frac{3}{2}}}) = 0

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

= \frac{0}{7}

規則を適用

\frac{0}{a} = 0, a \neq = 0

= 0