What is the limit as x approaches infinity of (e^x+e^{-x})/(3e^{-x)-e^x} ?

The limit as x approaches infinity of (e^x+e^{-x})/(3e^{-x)-e^x} is -1

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

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

- 1

Solution steps

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

Divide by

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

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

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

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

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

= \frac{1}{- 1}

Simplify

= - 1

n \to \infty lim (\frac{1}{3})

t \to 1 lim (\frac{5 t ^{4} - 4 t ^{2} - 1}{10 - t - 9 t ^{3}})

x \to \frac{π}{2} - lim (\frac{1}{cos ( x )})

x \to 2 + lim (3 x - 1)

n \to \infty lim (\frac{3 ^{2 n}}{n !})

What is the limit as x approaches infinity of (e^x+e^{-x})/(3e^{-x)-e^x} ?
The limit as x approaches infinity of (e^x+e^{-x})/(3e^{-x)-e^x} is -1