What is the limit as x approaches infinity of (cosh(x))/(e^x) ?

The limit as x approaches infinity of (cosh(x))/(e^x) is 1/2

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limit as x approaches infinity of (cosh(x))/(e^x)

x \to \infty lim (\frac{cosh ( x )}{e ^{x}})

\frac{1}{2}

Decimal Notation

0.5

Solution steps

x \to \infty lim (\frac{cosh ( x )}{e ^{x}})

Use the following identity:

cosh (x) = \frac{e ^{- x} + e ^{x}}{2}

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

x \to a lim [c \cdot f (x)] = c \cdot x \to a lim f (x)

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

Simplify

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

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

Divide by

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

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

x \to a lim [f (x) \pm g (x)] = x \to a lim f (x) \pm x \to a lim g (x)

With the exception of indeterminate form

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

x \to \infty lim (\frac{1}{e ^{2 x}}) = 0

x \to \infty lim (1) = 1

= \frac{1}{2} (0 + 1)

Simplify

= \frac{1}{2}

x \to 0 lim (sin^{2} (\frac{1}{x}))

x \to - 1 lim (3 x^{3} - 4 x^{2} - 3)

x \to 0 lim (e^{\frac{2}{x}})

x \to 2 lim (x^{3} + 2 x - 8)

x \to 0 lim (\frac{x - 3}{x ^{2} - 9})

What is the limit as x approaches infinity of (cosh(x))/(e^x) ?
The limit as x approaches infinity of (cosh(x))/(e^x) is 1/2