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

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

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

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

\frac{49}{9}

Decimal

5.44444 \dots

Solution steps

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

Apply L'Hopital's Rule

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

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

Apply L'Hopital's Rule

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

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

Plug in the value

x = 0

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

Simplify

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

= \frac{49}{9}

x \to 0 lim (x^{l n (x + 1)})

x \to - 1 lim (ln ∣ 1 + x ∣)

x \to 4 lim (4 - x^{2})

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

x \to - 2 lim (2 x - 3)

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