What is the limit as x approaches 0 of (sin(4x)cos(3x))/(7x) ?

The limit as x approaches 0 of (sin(4x)cos(3x))/(7x) is 4/7

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limit as x approaches 0 of (sin(4x)cos(3x))/(7x)

x \to 0 lim (\frac{sin ( 4 x ) cos ( 3 x )}{7 x})

\frac{4}{7}

Decimal

0.57142 \dots

Solution steps

x \to 0 lim (\frac{sin ( 4 x ) cos ( 3 x )}{7 x})

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

= \frac{1}{7} \cdot x \to 0 lim (\frac{sin ( 4 x ) cos ( 3 x )}{x})

Apply L'Hopital's Rule

= \frac{1}{7} \cdot x \to 0 lim (\frac{4 cos ( 4 x ) cos ( 3 x ) - 3 sin ( 3 x ) sin ( 4 x )}{1})

Plug in the value

x = 0

= \frac{1}{7} \cdot \frac{4 cos ( 4 \cdot 0 ) cos ( 3 \cdot 0 ) - 3 sin ( 3 \cdot 0 ) sin ( 4 \cdot 0 )}{1}

Simplify

\frac{1}{7} \cdot \frac{4 cos ( 4 \cdot 0 ) cos ( 3 \cdot 0 ) - 3 sin ( 3 \cdot 0 ) sin ( 4 \cdot 0 )}{1} : \frac{4}{7}

= \frac{4}{7}

What is the limit as x approaches 0 of (sin(4x)cos(3x))/(7x) ?
The limit as x approaches 0 of (sin(4x)cos(3x))/(7x) is 4/7