Q156 / 270 · NDA 2010 (Phase 2) · Mathematics

What is value of $\lim _{x \rightarrow 0} \frac{\cos (a x)-\cos (b x)}{x^2}$

  • A
    $a-b$
  • B
    $a+b$
  • Correct answer
    C
    $\frac{b^2-a^2}{2}$
  • D
    $\frac{b^2+a^2}{2}$

Solution

Using L'Hospital rule $\begin{aligned} & \lim _{x \rightarrow 0} \frac{-a \sin a x+b \sin b x}{-2 x} \\ & \lim _{x \rightarrow 0} \frac{-a^2 \cos a x+b^2 \cos b x}{2} \\ & =\frac{b^2-a^2}{2} \end{aligned}$

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

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