What is the limit as x approaches 0 of (1-cos(3x))/(1-cos(7x)) ?

The limit as x approaches 0 of (1-cos(3x))/(1-cos(7x)) is 9/49

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limit as x approaches 0 of (1-cos(3x))/(1-cos(7x))

x \to 0 lim (\frac{1 - cos ( 3 x )}{1 - cos ( 7 x )})

\frac{9}{49}

Decimal

0.18367 \dots

Solution steps

x \to 0 lim (\frac{1 - cos ( 3 x )}{1 - cos ( 7 x )})

Apply L'Hopital's Rule

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

Apply L'Hopital's Rule

= x \to 0 lim (\frac{9 cos ( 3 x )}{49 cos ( 7 x )})

Plug in the value

x = 0

= \frac{9 cos ( 3 \cdot 0 )}{49 cos ( 7 \cdot 0 )}

Simplify

\frac{9 cos ( 3 \cdot 0 )}{49 cos ( 7 \cdot 0 )} : \frac{9}{49}

= \frac{9}{49}

What is the limit as x approaches 0 of (1-cos(3x))/(1-cos(7x)) ?
The limit as x approaches 0 of (1-cos(3x))/(1-cos(7x)) is 9/49