What is the limit as x approaches 0 of (2-2cos(x))/(e^x-x-1) ?

The limit as x approaches 0 of (2-2cos(x))/(e^x-x-1) is 2

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limit as x approaches 0 of (2-2cos(x))/(e^x-x-1)

x \to 0 lim (\frac{2 - 2 cos ( x )}{e ^{x} - x - 1})

2

Solution steps

x \to 0 lim (\frac{2 - 2 cos ( x )}{e ^{x} - x - 1})

Factor out

2 cos (x)

from

2 - 2 cos (x) : 2 cos (x) (\frac{- 2}{- 2 cos ( x )} - 1)

= x \to 0 lim \frac{2 cos ( x ) ( \frac{- 2}{- 2 c o s ( x )} - 1 )}{e ^{x} - x - 1}

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

= 2 \cdot x \to 0 lim \frac{cos ( x ) ( \frac{- 2}{- 2 c o s ( x )} - 1 )}{e ^{x} - x - 1}

Simplify

\frac{cos ( x ) ( \frac{- 2}{- 2 c o s ( x )} - 1 )}{e ^{x} - x - 1} : \frac{1 - cos ( x )}{e ^{x} - x - 1}

= 2 \cdot x \to 0 lim (\frac{1 - cos ( x )}{e ^{x} - x - 1})

Multiply by the conjugate of

1 - cos (x) : \frac{sin ^{2} ( x )}{1 + cos ( x )}

= 2 \cdot x \to 0 lim \frac{\frac{s i n ^{2} ( x )}{1 + c o s ( x )}}{e ^{x} - x - 1}

Use the following identity:

sin^{2} (x) = 1 - cos^{2} (x)

= 2 \cdot x \to 0 lim \frac{\frac{1 - c o s ^{2} ( x )}{1 + c o s ( x )}}{e ^{x} - x - 1}

Simplify

\frac{\frac{1 - c o s ^{2} ( x )}{1 + c o s ( x )}}{e ^{x} - x - 1} : - \frac{cos ( x ) - 1}{- x + e ^{x} - 1}

= 2 \cdot x \to 0 lim (- \frac{cos ( x ) - 1}{- x + e ^{x} - 1})

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

= 2 (- x \to 0 lim (\frac{cos ( x ) - 1}{- x + e ^{x} - 1}))

Apply L'Hopital's Rule

: x \to 0 lim (\frac{- sin ( x )}{e ^{x} - 1})

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

Apply L'Hopital's Rule

: x \to 0 lim (\frac{- cos ( x )}{e ^{x}})

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

Plug in the value

x = 0

= 2 (- \frac{- cos ( 0 )}{e ^{0}})

Simplify

2 (- \frac{- cos ( 0 )}{e ^{0}}) : 2

= 2

\int \frac{( ln ( x ) )}{x ^{2}} d x

\frac{\partial}{\partial x} (x^{8} y^{6} - x^{7} y^{5})

x \to \infty lim (7^{- x})

\frac{\partial}{\partial x} (\frac{x + 6 y}{y ^{2} + 3 x ^{2}})

\int_{1}^{4} \frac{2}{x ^{2} + 4 x + 4} d x

What is the limit as x approaches 0 of (2-2cos(x))/(e^x-x-1) ?
The limit as x approaches 0 of (2-2cos(x))/(e^x-x-1) is 2