Question

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

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

Answer 1

Answer: (C)

Let $\frac{1}{r}=x$ so that as $r\to \infty ,x\to 0$
$\underset{r\to \infty }{\mathrm{Lim}}\frac{\cos ax-\cos bx\cdot \cos cx}{\frac{\sin bx}{bx}\cdot \frac{\sin cx}{cx}\cdot bc\cdot x^2}=\frac{1}{bc}\underset{r\to \infty }{\mathrm{Lim}}\frac{\cos ax-\cos bx\cdot \cos cx}{x^2}$
use L'Hospital's or expansion to get (C)