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

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

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

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

\frac{1}{2}

Decimal

0.5

Solution steps

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

\frac{sin ( x ) cos ( x )}{x ( 1 + cos ( x ) )} = \frac{sin ( x ) cos ( x )}{x} \cdot \frac{1}{( 1 + cos ( x ) )}

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

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

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

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

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

Simplify

= \frac{1}{2}

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