Limits and Deri...

If $\displaystyle \lim_{x\rightarrow \infty}\dfrac{x^3+1}{x^2+1}-(ax+b)=2$, then

The value of the constant $\alpha$ and $\beta$ such that $\displaystyle \lim_{x\rightarrow \infty}\left(\displaystyle\frac{x^2+1}{x+1}-\alpha x-\beta\right)=0$ are respectively.

$\displaystyle \lim_{x\rightarrow 0}\dfrac {1 - \cos x}{x^{2}}$ is ____

The limit of $\left[\frac{1}{x^2}+\frac{(2013)^x}{e^x-1}-\frac{1}{e^x-1}\right]$ as $x\rightarrow 0$

If the function $f(x)$ satisfies $\displaystyle \lim_{x\rightarrow 1}\frac{f(x)-2}{x^2-1}=\pi$, then $\displaystyle \lim_{x\rightarrow 1}f(x)=$

$f(x) = \log \left (e^{x} \left (\dfrac {x - 2}{x + 2}\right )^{\dfrac {3}{4}} \right ) \Rightarrow f'(0) =$

If $f(x) = \sec (3x)$, then $f'\left (\dfrac {3\pi}{4}\right ) =$

If $y = f(x^{2} + 2)$ and $f'(3) = 5$, then $\dfrac {dy}{dx}$ at $x = 1$ is _____

If f(x) is a function such that $f^{\prime \prime}(x)+f(x)=0$ and $g(x)=[f(x)]^2+[f'(x)]^2$ and g(3)=8, then $g(8)=$_____

The limit of $\displaystyle \sum_{n=1}^{1000}(-1)^nx^n$ as $x\rightarrow \infty$