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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Single Correct Medium Published on 17th 09, 2020
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