List of top Mathematics Questions on limits and derivatives asked in AIEEE

$\displaystyle \lim_{x\to0} \frac{sin\left(\pi\,cos^{2} x\right)}{x^{2}}$ equals
Let $\alpha$ and $\beta$ be the distinct roots of $ax^2 + bx + c = 0$ , then $\displaystyle \lim_{x \to\alpha} \frac{1- \cos \left(ax^{2} + bx + c\right)}{\left(x-\alpha\right)^{2}} $ is equal to
$\displaystyle \lim_{n \to\infty} \left[\frac{1}{n^{2}} \sec^{2} \frac{1}{n^{2}} + \frac{2}{n^{2}} \sec^{2} \frac{4}{n^{2}}.......... + \frac{1}{n}\sec^{2} 1 \right] $ equals
If $\displaystyle\lim_{x \to\infty} \left(1+ \frac{a}{x} + \frac{b}{x^{2}}\right)^{2x} = e^{2} $ , then the values of a and b, are
$\displaystyle\lim_{x \to 0} \frac{ \sqrt{1 -\cos \, 2x}}{\sqrt{2} x}$ is