What is the limit as x approaches 0 of (1-cos(8x))/(1-cos(3x)) ?

The limit as x approaches 0 of (1-cos(8x))/(1-cos(3x)) is 64/9

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limit as x approaches 0 of (1-cos(8x))/(1-cos(3x))

x \to 0 lim (\frac{1 - cos ( 8 x )}{1 - cos ( 3 x )})

\frac{64}{9}

Decimal

7.11111 \dots

Solution steps

x \to 0 lim (\frac{1 - cos ( 8 x )}{1 - cos ( 3 x )})

Apply L'Hopital's Rule

: x \to 0 lim (\frac{8 sin ( 8 x )}{3 sin ( 3 x )})

= x \to 0 lim (\frac{8 sin ( 8 x )}{3 sin ( 3 x )})

Apply L'Hopital's Rule

: x \to 0 lim (\frac{64 cos ( 8 x )}{9 cos ( 3 x )})

= x \to 0 lim (\frac{64 cos ( 8 x )}{9 cos ( 3 x )})

Plug in the value

x = 0

= \frac{64 cos ( 8 \cdot 0 )}{9 cos ( 3 \cdot 0 )}

Simplify

\frac{64 cos ( 8 \cdot 0 )}{9 cos ( 3 \cdot 0 )} : \frac{64}{9}

= \frac{64}{9}

x \to 3 lim ((x - 3)^{x - 3})

x \to 0 lim (e^{c o s (x)})

x \to x lim (e^{x})

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

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

What is the limit as x approaches 0 of (1-cos(8x))/(1-cos(3x)) ?
The limit as x approaches 0 of (1-cos(8x))/(1-cos(3x)) is 64/9