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 $$x\rightarrow \infty$$