12. limits — ગણિત ધોરણ 11 સાયન્સ — Question

$ = \mathop {\lim }\limits_{x \to 0} \,\frac{{2\,\sin \,\left( {\frac{{a + b}}{2}} \right)x\,.\,\sin \,\left( {\frac{{b - a}}{2}} \right)\,x}}{{\left( {\frac{{a + b}}{2}} \right)x\,.\frac{2}{{a + b}}.\frac{2}{{b - a}}.\left( {\frac{{b - a}}{2}} \right)x}} = \frac{{{b^2} - {a^2}}}{2}$

Aliter : Apply  $ L$-Hospital’s rule,

$\mathop {\lim }\limits_{x \to 0} \,\frac{{\cos ax - \cos bx}}{{{x^2}}} = \mathop {\lim }\limits_{x \to 0} \,\,\frac{{ - \,a\sin ax + b\sin bx}}{{2x}}$

$ = \mathop {\lim }\limits_{x \to 0} \,\frac{{ - \,{a^2}\cos ax + {b^2}\cos bx}}{2} = \frac{{{b^2} - {a^2}}}{2}.$