What is the limit as x approaches 0 of (x-sin(2x))/x ?

The limit as x approaches 0 of (x-sin(2x))/x is -1

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limit as x approaches 0 of (x-sin(2x))/x

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

- 1

Solution steps

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

Factor out

x

from

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

= x \to 0 lim \frac{x ( 1 - \frac{s i n ( 2 x )}{x} )}{x}

Cancel the common factor:

x

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

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

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

x \to 0 lim (1) = 1

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

= 1 - 2

Simplify

= - 1

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