Q100 / 160 · AP EAMCET 2024 (22 May Shift 2) · Mathematics

$\lim _{x \rightarrow 0} \frac{\cos 2 x-\cos 3 x}{\cos 4 x-\cos 5 x}=$

  • Correct answer
    A
    $\frac{5}{9}$
  • B
    $1$
  • C
    $\frac{3}{4}$
  • D
    $\frac{2}{5}$

Solution

$\begin{aligned} & \lim _{x \rightarrow 0} \frac{\cos 2 x-\cos 3 x}{\cos 4 x-\cos 5 x}=\lim _{x \rightarrow 0} \frac{-2 \sin 2 x+3 \sin 3 x}{-4 \sin 4 x+5 \sin 5 x} \\ & =\lim _{x \rightarrow 0} \frac{-4 \cos 2 x+9 \cos 3 x}{(-16) \cos 4 x+25 \cos 5 x}=\frac{-4+9}{-16+25}=\frac{5}{9}\end{aligned}$

Hence, the correct answer is option: 'A'.

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