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

The limit as x approaches 0 of (x-tan(x))/x is 0

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limit as x approaches 0 of (x-tan(x))/x

x \to 0 lim (\frac{x - tan ( x )}{x})

0

Solution steps

x \to 0 lim (\frac{x - tan ( x )}{x})

Factor out

x

from

x - tan (x) : x (1 - \frac{tan ( x )}{x})

= x \to 0 lim \frac{x ( 1 - \frac{t a n ( x )}{x} )}{x}

Cancel the common factor:

x

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

x \to 0 lim (1) = 1

x \to 0 lim (\frac{tan ( x )}{x}) = 1

= 1 - 1

Simplify

= 0

What is the limit as x approaches 0 of (x-tan(x))/x ?
The limit as x approaches 0 of (x-tan(x))/x is 0