নিচের সীমার মান কোনটি? $\lim _{x \rightarrow 0} \frac{\mathrm{e}^{x^{2}}-\cos x}{\mathrm{x}^{2}}$

$\begin{array}{l} =\lim _{x \rightarrow 0} \frac{1}{x^{2}}\left\{\left(1+\frac{x^{2}}{1 !}+\frac{\left(x^{2}\right)^{2}}{2 !}+\cdots \infty\right)-\right. \\ \left.\left(1-\frac{x^{2}}{2 !}+\frac{x^{4}}{4 !}-\cdots \infty\right)\right\} \\ \left.=\lim _{x \rightarrow 0} \frac{1}{x^{2}}\left\{\left(\frac{1}{1 !}+\frac{1}{2 !}\right) x^{2}+\left(\frac{1}{2 !}-\frac{1}{4 !}\right) x^{4}+\cdots \infty\right)\right\} \\ =\lim _{x \rightarrow 0}\left\{\frac{3}{2}+\left(\frac{1}{2 !}-\frac{1}{4 !}\right) x^{2}+x\right. \text { এর উচ্চঘাত সম্বলিত } \end{array}$

$=\frac{3}{2}+0=\frac{3}{2}(\mathrm{Ans})$

Alternate: $\lim _{x \rightarrow 0} \frac{e^{x^{2}}-\cos x}{x^{2}}$

$\begin{array}{l} =\lim _{x \rightarrow 0} \frac{e^{x^{2}}-1+1-\cos x}{x^{2}} \\ =\lim _{x \rightarrow 0} \frac{e^{x^{2}}-1}{x^{2}}++_{x \rightarrow 0} \frac{1-\cos x}{x^{2}} \\ =1+\lim _{x \rightarrow 0} \frac{2 \sin _{f}^{2}(x / 2)}{x^{2}} \\ =1+2 \times \lim _{x \rightarrow 0} \frac{\sin ^{2}(x / 2)}{(x / 2)^{2}} \times \frac{1}{4} \\ =1+1 \times \frac{1}{2}=\frac{3}{2} \end{array}$