Question

$ \displaystyle \lim_{x \to 0} $ $ \frac{cos\,ax-cos\,bx}{x^{2}} $ is equal to

• A. $ \frac{a^{2}-b^{2}}{2} $
• B. $ \frac{b^{2}-a^{2}}{2} $
• C. $ a^2 - b^2 $
• D. $ b^2 - a^2 $

Answer 1

Answer: (B)

$\displaystyle\lim _{x \rightarrow 0} \frac{\cos a x-\cos b x}{x^{2}}$
$= \displaystyle\lim _{x \rightarrow 0} \frac{-a \sin a x+b \sin b x}{2 x}$
(using L'Hospital's rule)
$= \displaystyle\lim _{x \rightarrow 0} \frac{-a^{2} \cos a x+b^{2} \sin b x}{2}$
(using L'Hospital's rule)
$=\frac{b^{2}-a^{2}}{2}$