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

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

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

2

Solution steps

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

Divide by

e^{j} : \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

With the exception of indeterminate form

= \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}

Simplify

= 2

y = sec^{2} (x)

n \to \infty lim (i = 1 \sum n \frac{2 i}{n} \cdot \frac{2}{n})

\frac{\partial}{\partial x} (\frac{1}{x + 2})

f (x) = 3 x - 4 x^{2}, (- 1, - 7)

\int 5 x x + 4 d x

What is the limit as j approaches infinity of (2e^j+e^{-j}+2e^4+1)/(e^j+e^4) ?
The limit as j approaches infinity of (2e^j+e^{-j}+2e^4+1)/(e^j+e^4) is 2