If $f(x) = \lim_{n \rightarrow \infty} \left( \frac{\log(2+x) - x^{2n} \sin x}{1+x^{2n}} \right)$ for $0 \leq x \leq \frac{\pi}{2}$, then at $x=1$, $f(x)$ is

Let $x=2$ be a root of the equation $x^2+px+q=0$ and $f(x)=\begin{cases} \frac{1-\cos(x^2-4px+q^2+8q+16)}{(x-2p)^4}, & x \neq 2p \\ 0, & x=2p \end{cases}$. Then $\lim _{x \rightarrow 2p^{+}}[f(x)]$,where $[.]$ denotes the greatest integer function,is $........$

If $f(x) = \begin{cases} x^2, & x \ne 1 \\ 2, & x = 1 \end{cases}$,then which of the following is true?

Let $f: R \rightarrow R$ be a function given by $f(x) = \begin{cases} \frac{1-\cos 2x}{x^2} & , x < 0 \\ \alpha & , x = 0 \\ \frac{\beta \sqrt{1-\cos x}}{x} & , x > 0 \end{cases}$. If $f$ is continuous at $x = 0$,then $\alpha^2 + \beta^2$ is equal to:

The function $f(x) = \frac{x^2 - 1}{x^3 - 1}$ is not defined at $x = 1$. If the function is continuous at $x = 1$,then the value of $f(1)$ will be:

Let $f(x) = \begin{cases} (x - 1)^{\frac{1}{2 - x}}, & x > 1, x \neq 2 \\ k, & x = 2 \end{cases}$. The value of $k$ for which $f$ is continuous at $x = 2$ is