What is the limit as x approaches-infinity of (6x)/(x+cos(x)) ?

The limit as x approaches-infinity of (6x)/(x+cos(x)) is 6

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limit as x approaches-infinity of (6x)/(x+cos(x))

x \to - \infty lim (\frac{6 x}{x + cos ( x )})

6

Solution steps

x \to - \infty lim (\frac{6 x}{x + cos ( x )})

x \to a lim [c \cdot f (x)] = c \cdot x \to a lim f (x)

= 6 \cdot x \to - \infty lim (\frac{x}{x + cos ( x )})

Divide by highest denominator power:

\frac{1}{1 + \frac{c o s ( x )}{x}}

= 6 \cdot x \to - \infty lim (\frac{1}{1 + \frac{c o s ( x )}{x}})

x \to a lim [\frac{f ( x )}{g ( x )}] = \frac{lim _{x \to a} f ( x )}{lim _{x \to a} g ( x )}, x \to a lim g (x) \neq = 0

With the exception of indeterminate form

= 6 \cdot \frac{lim _{x \to - \infty} ( 1 )}{lim _{x \to - \infty} ( 1 + \frac{c o s ( x )}{x} )}

x \to - \infty lim (1) = 1

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

= 6 \cdot \frac{1}{1}

Simplify

= 6

x \to 3 lim (x^{2} - 6 x + 4)

x \to 0 lim (\frac{5 x - tan ( 5 x )}{x ^{3}})

x \to - 7 lim (\frac{x ^{2} + 7 x}{x ^{2} - 2 x - 63})

x \to 0 lim (csc (x) sin (sin (x)))

t \to \infty lim (\frac{ln ( 3 t )}{t ^{2}})

What is the limit as x approaches-infinity of (6x)/(x+cos(x)) ?
The limit as x approaches-infinity of (6x)/(x+cos(x)) is 6