12. limits — ગણિત ધોરણ 11 સાયન્સ — Question

Now expanding ${e^{{x^2}}}$ and $\cos x,$ we get

$\mathop {\lim }\limits_{x \to 0} \,\frac{{\frac{{3{x^2}}}{{2\,!}} + {x^4}\,\left( {\frac{1}{{2\,!}} - \frac{1}{{4\,!}}} \right) + .......}}{{{x^2}}} = \frac{3}{2}$

Aliter : Apply $L-$ Hospital’s rule,

$\mathop {\lim }\limits_{x \to 0} \,\frac{{2x{e^{{x^2}}} + \sin x}}{{2x}} = \mathop {\lim }\limits_{x \to 0} \,{e^{{x^2}}} + \mathop {\lim }\limits_{x \to 0} \,\frac{{\sin x}}{{2x}} $

$= 1 + \frac{1}{2} = \frac{3}{2}.$