What is the limit as x approaches infinity of (x+3cos(x))/x ?

The limit as x approaches infinity of (x+3cos(x))/x is 1

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limit as x approaches infinity of (x+3cos(x))/x

x \to \infty lim (\frac{x + 3 cos ( x )}{x})

1

Solution steps

x \to \infty lim (\frac{x + 3 cos ( x )}{x})

Factor out

x

from

x + 3 cos (x) : x (1 + \frac{3 cos ( x )}{x})

= x \to \infty lim \frac{x ( 1 + \frac{3 c o s ( x )}{x} )}{x}

Cancel the common factor:

x

= x \to \infty lim (1 + \frac{3 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{3 cos ( x )}{x})

x \to \infty lim (1) = 1

x \to \infty lim (\frac{3 cos ( x )}{x}) = 0

= 1 + 0

Simplify

= 1

x \to 0 lim (\frac{cos ( x )}{3})

x \to 1 - lim (arccos (x))

x \to - \infty lim (3 x - x^{3})

x \to 0 lim (- 2 x + x^{4} - x^{2} - 1)

x \to - \infty lim (x^{2} + 6 x - x)

What is the limit as x approaches infinity of (x+3cos(x))/x ?
The limit as x approaches infinity of (x+3cos(x))/x is 1