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

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

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

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

2

Solution steps

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

Apply L'Hopital's Rule

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

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

Plug in the value

x = 0

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

Simplify

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

= 2

\frac{d}{d x} (\frac{- 3 x ^{4} - 5 x ^{3} + 2}{- 5 x ^{4}})

\frac{\partial}{\partial x} (\frac{1}{1 + x})

\frac{\partial}{\partial x} (x^{2} - 2 x^{2} y - 5 x y^{2})

\int (2 x)^{2} (5 + 3^{4})^{\frac{1}{4}} d x

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