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

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

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

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

1

Solution steps

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

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

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

Apply L'Hopital's Rule

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

Plug in the value

x = 0

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

Simplify

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

= 1

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