Mathematics

If f(x) $$If(x)=\begin{cases} \frac { { 36 }^{ x }-{ 9 }^{ x }-4^{ x }+1 }{ \sqrt { 2 } -\sqrt { 1+cosx }  } ,x\neq 0 \\ \quad \quad \quad \quad k\quad \quad \quad \quad \quad \quad \quad \quad \quad ,x=0 \end{cases}$$ is continuous at x=0, then k equals


ANSWER

$$16\sqrt { 2 }$$ In $$2$$ In $$3$$


SOLUTION
$$f\left( x \right) =\left\{ \begin{array}{l} \dfrac { { { { 36 }^{ x } }-{ 9^{ x } }-{ 4^{ x } } } }{ { \sqrt { 2 } -\sqrt { 1+\cos  x }  } } +1\, \, \, \, ,x\ne 0 \\ k\, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, x=0 \end{array} \right.  \\ { \lim   }_{ x\to 0 }\dfrac { { { 9^{ x } }\left( { { 4^{ x } }-1 } \right) -\left( { { 4^{ x } }-1 } \right)  } }{ { \sqrt { 2 } -\sqrt { 1+\cos  x }  } }  \\ { \lim   }_{ x\to 0 }\dfrac { { \left( { { 9^{ x } }-1 } \right) \left( { { 4^{ x } }-1 } \right) \left( { \sqrt { 2 } +\sqrt { 1+\cos  x }  } \right)  } }{ { \left( { 1-\cos  x } \right)  } }  \\ 2\sqrt { 2 } { \lim   }_{ x\to 0 }\dfrac { { \left( { { 9^{ x } }-1 } \right) \left( { { 4^{ x } }-1 } \right)  } }{ { \dfrac { { \left( { 1-\cos  x } \right)  } }{ { { x^{ 2 } } } } { x^{ 2 } } } }  \\ 4\sqrt { 2 } { \lim   }_{ x\to 0 }\, \, \dfrac { { \left( { { 9^{ x } }-1 } \right)  } }{ x } \dfrac { { \left( { { 4^{ x } }-1 } \right)  } }{ x }  \\ f\left( 0 \right) =4\sqrt { 2 } \, \ln9\, \ln4 \\ f\left( 0 \right) =16\sqrt { 2 } \, \, In\, \, \, 3\, \, In\, \, 2$$

Option $$C$$ is correct.
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