What is the limit as x approaches infinity of (x-cos(x))/x ?

The limit as x approaches infinity of (x-cos(x))/x is 1

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limit as x approaches infinity of (x-cos(x))/x

x \to \infty lim (\frac{x - cos ( x )}{x})

1

Solution steps

x \to \infty lim (\frac{x - cos ( x )}{x})

Divide by highest denominator power:

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

= x \to \infty lim (1 - \frac{cos ( 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 \infty lim (1) - x \to \infty lim (\frac{cos ( x )}{x})

x \to \infty lim (1) = 1

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

= 1 - 0

Simplify

= 1

What is the limit as x approaches infinity of (x-cos(x))/x ?
The limit as x approaches infinity of (x-cos(x))/x is 1