Question

Let $a , b , c$ are non zero constant number then $\underset{ r \rightarrow \infty}{\text{Lim}} \frac{\cos \frac{ a }{ r }-\cos \frac{ b }{ r } \cos \frac{ c }{ r }}{\sin \frac{ b }{ r } \sin \frac{ c }{ r }}$ equals

• A. $\frac{a^2+b^2-c^2}{2 b c}$
• B. $\frac{ c ^2+ a ^2- b ^2}{2 bc }$
• C. $\frac{ b ^2+ c ^2- a ^2}{2 bc }$
• D. independent of a, b and c

Answer 1

Answer: (C)

Let $\frac{1}{r}=x$ so that as $r \rightarrow \infty, x \rightarrow 0$
$\underset{ r \rightarrow \infty}{\text{Lim}} \frac{\cos a x-\cos b x \cdot \cos c x}{\frac{\sin b x}{b x} \cdot \frac{\sin c x}{c x} \cdot b c \cdot x^2}=\frac{1}{b c}\underset{ r \rightarrow \infty}{\text{Lim}} \frac{\cos a x-\cos b x \cdot \cos c x}{x^2}$
use L'Hospital's or expansion to get (C)