limit as x approaches infinity of (1-5e^{2x})/(3e^x+4e^{2x)}

解

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

- \frac{5}{4}

十進法表記

- 1.25

解答ステップ

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

e^{2 x}

で割る

: \frac{\frac{1}{e ^{2 x}} - 5}{\frac{3}{e ^{x}} + 4}

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

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

x \to \infty lim (\frac{1}{e ^{2 x}} - 5) = - 5

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

= \frac{- 5}{4}

簡素化

= - \frac{5}{4}