12. limits — MATHS STD 11 Science — Question

$ = \mathop {\lim }\limits_{y \to 0} \,\left[ {\frac{x}{y}\,\left\{ {\frac{{\cos x - \cos \,(x + y)}}{{\cos \,(x + y)\,\cos x}}} \right\}} \right] + \mathop {\lim }\limits_{y \to 0} \sec \,(x + y)$

$ = \mathop {\lim }\limits_{y \to 0} \,\left[ {\frac{{x\sin \,\left( {x + \frac{y}{2}} \right)}}{{\cos \,(x + y)\,.\,\,\cos x}}\,.\,\frac{{\sin \,\left( {\frac{y}{2}} \right)}}{{\,\,\,\frac{y}{2}}}} \right] + \sec x$

$= x\ tan\ x\ sec\ x + sec\ x = sec\ x\ (x\ tan\ x+1).$

Aliter : Apply $L-$ Hospital’s rule,

$\mathop {\lim }\limits_{y \to 0} \,\frac{{(x + y)\,\sec \,(x + y) - x\,\sec x}}{y}$

$ = \mathop {\lim }\limits_{y \to 0} \,\frac{{(x + y)\,\sec \,(x + y)\tan \,(x + y) + \sec \,(x + y) - 0}}{1}$

{Differentiating w.r.t.$y$ assuming $x$ as constant}

$ = x\sec x\tan x + \sec x. = \sec x(x\tan x + 1)$