Limits and Cont...

If $$\displaystyle \lim _{ n\rightarrow \infty  }{ \cfrac { n.{ 3 }^{ n } }{ n{ \left( x-2 \right)  }^{ n }+n.{ 3 }^{ n+1 }-{ 3 }^{ n } }  } =\cfrac { 1 }{ 3 } $$, then the range of $$x$$ is (When $$n\in N$$)

$$\displaystyle \lim_{x\rightarrow 3} = \dfrac {\sqrt {x} -\sqrt {3}}{\sqrt {x^{2} - 9}}$$ is equal to

What is $$\displaystyle \lim_{x \rightarrow 0 }  x^2 \sin \left(\frac{1}{x}\right)$$ equal to ? 

If $$f\left( x \right) =\begin{vmatrix} \sin { x }  & \cos { x }  & \tan { x }  \\ { x }^{ 3 } & { x }^{ 2 } & x \\ 2x & 1 & 1 \end{vmatrix}$$, then $$\displaystyle\lim _{ x\rightarrow 0 }{ \dfrac { f\left( x \right)  }{ { x }^{ 2 } }  } $$ is

The value of $$\displaystyle \lim _{ x\rightarrow \pi /6 }{ \cfrac { 2\sin ^{ 2 }{ x } +\sin { x } -1 }{ 2\sin ^{ 2 }{ x } -3\sin { x } -1 }  } $$ is

Which one of the following statements is correct?

$$\displaystyle\lim_{x\rightarrow \frac{\pi}{2}}(\pi - 2x^{\cos x})$$ is equal to :

$$\displaystyle\lim_{x\rightarrow\frac{\pi}{6}}\frac{\sin\left(x-\displaystyle\frac{\pi}{6}\right)}{\sqrt{3-2cos x}}$$ is equal to :

If $$\underset{x\to 0}{\lim}\dfrac{x^a\sin^b x}{\sin(x^c)}, a, b, c, \in R \sim \{0\}$$ exists and has non-zero value, then 

Determine the value of k for which the following function is continuous at $$x=3$$.