What is the limit as x approaches 0 of (x-sin(x)}{e^x-1-x-\frac{x^2)/2} ?

The limit as x approaches 0 of (x-sin(x)}{e^x-1-x-\frac{x^2)/2} is 1

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limit as x approaches 0 of (x-sin(x)}{e^x-1-x-\frac{x^2)/2}

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

1

Solution steps

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

Apply L'Hopital's Rule

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

Apply L'Hopital's Rule

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

Apply L'Hopital's Rule

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

Plug in the value

x = 0

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

Simplify

\frac{cos ( 0 )}{e ^{0}} : 1

= 1

What is the limit as x approaches 0 of (x-sin(x)}{e^x-1-x-\frac{x^2)/2} ?
The limit as x approaches 0 of (x-sin(x)}{e^x-1-x-\frac{x^2)/2} is 1