Let

$$\begin{aligned} f(x) & =x+a \sqrt{2} \sin x & & , 0 \leq x<\frac{\pi}{4} \\ & =2 x \cot x+b & & \frac{\pi}{4} \leq x<\frac{\pi}{2} \\ & =a \cos 2 x-b \sin x & & \frac{\pi}{2} \leq x \leq \pi \end{aligned}$$

If $$\mathrm{f}(\mathrm{x})$$ is continuous for $$0 \leq \mathrm{x} \leq \pi$$, then

$$\lim _\limits{x \rightarrow 2}(x-1)^{ \frac{1}{3 x-6}}=$$

$$\begin{aligned} & \text { } f(x)=\frac{\sqrt{1+p x}-\sqrt{1-p x}}{x} \text {, if } 1 \leq x<0 \\ & =\frac{2 x+1}{x-2} \quad \text {, if } 0 \leq x \leq 1 \\ \end{aligned}$$

is continuous in the interval $$[-1,1]$$, then $$p=$$

If $$\lim _\limits{x \rightarrow 5} \frac{x^k-5^k}{x-5}=500$$, then the value of $$k$$, where $$k \in N$$ is