Limits and Cont...

If $$f(x)=(x)^{\tfrac{1}{x-1}}$$ for $$x\neq 1$$ and $$\mathrm{f}$$ is continuous at $$\mathrm{x}=1$$ then $$\mathrm{f}(1)=$$

Assertion (A): $$f(x)=\displaystyle \frac{\sin\{[x]\pi\}}{1+x^{2}}$$ is continuous on $$\mathrm{R}$$ (where $$[x]$$ denotes greatest integer function of $$x $$).
Reason (R): Every constant function is continuous on $$\mathrm{R}$$

The values of $$p$$ and $$q$$ for which the function $$\mathrm{f}(\mathrm{x}) = \left\{\begin{array}{ll}
\dfrac{\sin(\mathrm{p}+1)\mathrm{x}+\sin \mathrm{x}}{\mathrm{x}} & , \mathrm{x}<0\\
\mathrm{q} & , \mathrm{x}=0\\
\dfrac{\sqrt{\mathrm{x}+\mathrm{x}^{2}}-\sqrt{\mathrm{x}}}{\mathrm{x}^{3/2}} & , \mathrm{x}>0
\end{array}\right.$$
is continuous for all $$\mathrm{x}$$ in $$\mathrm{R}$$, are

If $$\displaystyle f\left( x \right)=\begin{cases} \begin{matrix} x+\lambda , & -1<x<3 \end{matrix} \\ \begin{matrix} 4, & x=3 \end{matrix} \\ \begin{matrix} 3x-5, & x>3 \end{matrix} \end{cases}$$ is continuous at $$x=3$$ then tha value of $$\lambda$$ is

The function $$\displaystyle \mathrm{f}({x})=\begin{cases}\dfrac{1-\sin x}{(\pi-2x)^{2}} & x  \neq\dfrac{\pi}{2}\\ \mathrm{k}& {x}=\dfrac{\pi}{2}\end{cases}$$ is continuous at $$\displaystyle {x}=\dfrac{\pi}{2}$$ then $$\mathrm{k}$$ is equal to

The value of $$f(0)$$ so that the function $$\mathrm{f}({x})$$$$=\displaystyle \frac{1-\cos(1-\cos x)}{x^{4}}$$ is continuous everywhere is

The value of $$p$$ for which the function

$$f(x)=\displaystyle \left\{ \begin{array}{rl} \dfrac { (4^{ { x } }-1)^{ 3 } }{ \sin\dfrac { { x } }{ { p } } \log(1+\dfrac { { x }^{ 2 } }{ 3 } ) }  & { ;\quad x\neq 0 } \\ 12(\log  4)^{ 3 } & { ;\quad x=0\quad  } \end{array} \right. $$ is continuous at $${x}=0$$, is

If $$f(x)=\displaystyle \frac { 1 }{ x(3x+1) } $$ then at $${x}=0,\mathrm{f}({x})$$ is

Assertion(A):
$$f(x)=x(\displaystyle \frac{1+e^{1/x}}{1-e^{1/x}})(x\neq 0)$$ , $${f}(0)=0$$ is continuous at $${x}=0$$.
Reason(R) A function is said to be continuous at $$a$$ if both limits are exists and equal to $$\mathrm{f}({a})$$ .

If   $$f:R\rightarrow R$$   is a function defined by   $$ f(x)=[x]\displaystyle \cos\left(\frac{2x-1}{2}\pi\right)$$,   where   $$[{x}]$$   denotes the greatest integer function, then   $${f}$$   is: