What is the limit as x approaches 0+of 1/x sin(x/2) ?

The limit as x approaches 0+of 1/x sin(x/2) is 1/2

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limit as x approaches 0+of 1/x sin(x/2)

x \to 0 + lim (\frac{1}{x} sin (\frac{x}{2}))

\frac{1}{2}

Decimal

0.5

Solution steps

x \to 0 + lim (\frac{1}{x} sin (\frac{x}{2}))

Simplify

\frac{1}{x} sin (\frac{x}{2}) : \frac{sin ( \frac{x}{2} )}{x}

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

Apply L'Hopital's Rule

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

Plug in the value

x = 0

= \frac{cos ( \frac{0}{2} ) \frac{1}{2}}{1}

Simplify

\frac{cos ( \frac{0}{2} ) \frac{1}{2}}{1} : \frac{1}{2}

= \frac{1}{2}

What is the limit as x approaches 0+of 1/x sin(x/2) ?
The limit as x approaches 0+of 1/x sin(x/2) is 1/2