Limits and Cont...

If $$|x| < 1$$, then $$\displaystyle \lim_{n \rightarrow \infty }\{ (1 + x) (1+x^2)(1 + x^4) ..... (1 + x^{2n}) \}$$ is equal to

The value of $$\displaystyle\lim_{x\rightarrow\infty}{\frac{\cot^{-1}{(x^{-a}\log_a{x})}}{\sec^{-1}{a^x\log_x{a}}}}$$ for $$(a>1)$$ is equal to?

Let $$f(x)=\sin x$$, $$g(x)=\left [ x+1 \right ]$$ and $$g(f(x))=h(x)$$, where [.] is the greatest integer function. Then $$h^+\left ( \displaystyle \dfrac{\pi }{2} \right )$$ is

Which one of the following statement is true?

$$\displaystyle \lim_{n\to\infty }\frac{n^{p}\sin ^{2}\left ( n! \right )}{n+1}$$, $$0<p<1$$, is equal to

$$f\left( x \right)=\begin{cases} \sin { x } \qquad ;\qquad x\neq n\pi ,n=0,\pm 1,\pm 2,\pm 3..... \\ 2\qquad \qquad ;\qquad otherwise \end{cases}$$ and $$g\left( x \right) =\begin{cases} { x }^{ 2 }+1\qquad ;\qquad x\neq 0 \\ 4\qquad \qquad ;\qquad x=0 \end{cases}.$$ 

Which one of the following statements is true?

If $$\displaystyle \lim_{x\rightarrow 0}(f(x)\:g(x))$$ exists for any functions $$f$$ and $$g$$ then

If $$p\left( x \right) ={ a }_{ 0 }+{ a }_{ 1 }x+...+{ a }_{ n }{ x }^{ n }$$ and $$\left| p\left( x \right)  \right| \le \left| { e }^{ x-1 }-1 \right| $$ for all $$x\ge 0,$$ then $$\left| { a }_{ 1 }+2{ a }_{ 2 }+3{ a }_{ 3 }+...+n{ a }_{ n } \right| $$