$$\mathop {Lim}\limits_{x \to \infty } \,x\sin {\raise0.5ex\hbox{$\scriptstyle 1$} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle x$}} = \_\_\_\_\_.$$

If $$f(0)=2$$ and $$f'\left( x \right) = {1 \over {5 - {x^2}}},$$ then the lower and upper bounds of $$f(1)$$ estimated by the mean value theorem are ______.

If at every point of a certain curve , the slope of the tangent equals $${{ - 2x} \over y},$$ the curve is _______.

Consider the chopper circuit of figure. The chopper operates at $$400$$ $$Hz$$ and $$50$$% duty cycle. The load current remains almost ripple free at $$10A.$$ Assuming the input voltage to be $$200V$$ and the devices to be ideal, the turn-off time available to the thyristor $$Th$$ $$M$$ is____________ $$\mu s$$