Evaluate \(\lim _{ x\rightarrow \sqrt { 2 } }{ \frac { { x }^{ 9 }-3{ x }^{ 8 }+{ x }^{ 6 }-9{ x }^{ 4 }-4{ x }^{ 2 }-16x+84 }{ { x }^{ 5 }-3{ x }^{ 4 }-4x+12 } } \)

Find k if \(\lim _{ x\rightarrow 1 }{ \frac { { x }^{ 4 }-1 }{ x-1 } } =\lim _{ x\rightarrow k }{ \left( \frac { { x }^{ 3 }-{ k }^{ 3 } }{ { x }^{ 2 }-{ k }^{ 2 } } \right) } \)

Evaluate \(\lim _{ x\rightarrow \frac { \pi }{ 2 } }{ \left( \frac { \pi }{ 2 } -x \right) \tan { x } } \)

Examine the continuity of \(f\left( x \right) \quad at\quad x=\frac { 1 }{ 2 } where\quad f\left( x \right) =\begin{cases} \frac { 1 }{ 2 } -x,\quad 0\le x\le \frac { 1 }{ 2 } \quad \\ 1\quad ,\quad \quad x=\frac { 1 }{ 2 } \\ \frac { 3 }{ 2 } -x,\quad \frac { 1 }{ 2 }

If \(f\left( x \right) =\begin{cases} 1,\quad \quad x\le 3 \\ ax+b,\quad 3 is continuous, prove that a = 3 and b = - 8.