What is the limit as x approaches 0+of ((sqrt(x)cos(x)))/(2x) ?

The limit as x approaches 0+of ((sqrt(x)cos(x)))/(2x) is infinity

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limit as x approaches 0+of ((sqrt(x)cos(x)))/(2x)

x \to 0 + lim (\frac{( x cos ( x ) )}{2 x})

\infty

Solution steps

x \to 0 + lim (\frac{x cos ( x )}{2 x})

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

= \frac{1}{2} \cdot x \to 0 + lim (\frac{x cos ( x )}{x})

Simplify

\frac{x cos ( x )}{x} : \frac{cos ( x )}{x}

= \frac{1}{2} \cdot x \to 0 + lim (\frac{cos ( x )}{x})

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

= \frac{1}{2} \cdot x \to 0 + lim (cos (x) \frac{1}{x})

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

With the exception of indeterminate form

= \frac{1}{2} \cdot x \to 0 + lim (cos (x)) \cdot x \to 0 + lim (\frac{1}{x})

x \to 0 + lim (cos (x)) = 1

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

= \frac{1}{2} \cdot 1 \cdot \infty

Apply Infinity Property:

c \cdot \infty = \infty

= \infty

What is the limit as x approaches 0+of ((sqrt(x)cos(x)))/(2x) ?
The limit as x approaches 0+of ((sqrt(x)cos(x)))/(2x) is infinity