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

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

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

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

0

Solution steps

x \to 0 lim (\frac{tan ( x ) - sin ( 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{tan ( x ) - sin ( x )}{x})

Apply L'Hopital's Rule

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

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

Plug in the value

x = 0

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

Simplify

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

= 0

x \to 0 lim (\frac{x ^{3} - 8 x}{x ^{2} + 4 x})

x \to 2 lim (7 x)

x \to 4 lim (\frac{0}{0})

x \to - 5 lim (\frac{x + 5}{∣ x + 5 ∣})

x \to 2 lim (6 x)

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