Evaluate the following limit : lim(x→0) (1 - cos 2x)/(cos 2x - cos 8x)

Question

Evaluate the following limit : \(\lim\limits_{\text x \to0}\cfrac{1-cos\,2\text x}{cos\,2\text x-cos\,8\text x} \)

lim(x→0) (1 - cos 2x)/(cos 2x - cos 8x)

Answer 1

To find: \(\lim\limits_{\text x \to0}\cfrac{1-cos\,2\text x}{cos\,2\text x-cos\,8\text x} \)

We know,

cos2x = 1 – 2sin²x

⇒ 2sin²x = 1 – cos2x

Dividing numerator and denominator by x²:

Put 3x = y & 5x = t:

Hence, the value of \(\lim\limits_{\text x \to0}\cfrac{1-cos\,2\text x}{cos\,2\text x-cos\,8\text x} \) = \(\cfrac1{15}\)