Limits and Cont...

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.

The function represented by  the following graph is.

If [x] denotes the greatest integer not exceeding $$x$$ and if the function $$f$$ defined by $$f(x)= \begin{cases}\dfrac{a+2\cos\,x}{x^2}&(x < 0) \\ b\,\tan \dfrac{\pi}{[x+4]}&(x \ge 0) \end{cases}$$ is continuous at $$x=0$$, then the order pair (a, b) =

$$\displaystyle \lim_{x\rightarrow 0}\dfrac {1 - \cos x}{x^{2}}$$ 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 $$\displaystyle \lim_{x\rightarrow 1}f(x)=$$

$$\displaystyle \lim_{x\rightarrow 0}\frac{log_e(1+x)}{3^x-1}=$$ ____.

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

The limit of $$\left\{\frac{1}{x}\sqrt{1-x}-\sqrt{1+\frac{1}{x^2}}\right\}$$ as $$x\rightarrow 0$$

$$\displaystyle \lim_{x\rightarrow \pi/4} \dfrac {\tan x - 1}{\cos 2x}$$ is equal to