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The value of lim x→0(1-cos 3x)/(1-cos 5x) is Single correct option
The value of
lim x → 0 1 − cos 3 x 1 − cos 5 x \displaystyle\lim_{x\to0}\dfrac{1-\cos3x}{1-\cos5x} x → 0 lim 1 − cos 5 x 1 − cos 3 x is
A. 3 5 \dfrac35 5 3 B. 25 9 \dfrac{25}9 9 25 C. 5 3 \dfrac53 3 5 D. 9 25 \dfrac9{25} 25 9 Show answer and solution Answer: (D) 9 25 \dfrac9{25} 25 9
Use
1 − cos k x ≈ k 2 x 2 2 1-\cos kx\approx\frac{k^2x^2}2 1 − cos k x ≈ 2 k 2 x 2 as
x → 0 x\to0 x → 0 . The ratio becomes
9 x 2 / 2 25 x 2 / 2 = 9 25 \frac{9x^2/2}{25x^2/2}=\frac9{25} 25 x 2 /2 9 x 2 /2 = 25 9 . The value
3 5 \frac35 5 3 forgets to square the coefficients, and
25 9 \frac{25}9 9 25 and
5 3 \frac53 3 5 invert the ratio. Equivalently,
2 sin 2 ( 3 x / 2 ) 2 sin 2 ( 5 x / 2 ) \frac{2\sin^2(3x/2)}{2\sin^2(5x/2)} 2 s i n 2 ( 5 x /2 ) 2 s i n 2 ( 3 x /2 ) gives the same.
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