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

解

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

2

解答ステップ

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

e^{j}

で割る

: \frac{2 + \frac{1}{e ^{2 j}} + 2 e ^{4 - j} + \frac{1}{e ^{j}}}{1 + e ^{4 - j}}

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

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

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

j \to \infty lim (1 + e^{4 - j}) = 1

= \frac{2}{1}

簡素化

= 2