Question

Evaluate the limit .​

Answer 1

Step-by-step explanation:

We have,

$\tt{ \lim_{ x \to \sqrt{2} } \: \dfrac{ {x}^{4} - 4 }{ {x}^{2} + 3x \sqrt{2} - 8 } }$

$\sf{ = \lim_{ x \to \sqrt{2} } \: \dfrac{ ({x}^{2} - 4 )( {x}^{2} + 4)}{ {x}^{2} + 4x \sqrt{2} - x \sqrt{2} - 8 } }$

$\sf{ = \lim_{ x \to \sqrt{2} } \: \dfrac{ (x - \sqrt{2} )(x + \sqrt{2}) ( {x}^{2} + 4)}{ x(x + 4\sqrt{2}) - \sqrt{2}(x + 4 \sqrt{2}) } }$

$\sf{ = \lim_{ x \to \sqrt{2} } \: \dfrac{ (x - \sqrt{2} )(x + \sqrt{2}) ( {x}^{2} + 4)}{ (x - \sqrt{2}) (x + 4\sqrt{2}) } }$

$\sf{ = \lim_{ x \to \sqrt{2} } \: \dfrac{ (x + \sqrt{2}) ( {x}^{2} + 4)}{ x + 4\sqrt{2} } }$

$\sf{ = \dfrac{ ( \sqrt{2} + \sqrt{2}) ( ( \sqrt{2} )^{2} + 4)}{ \sqrt{2} + 4\sqrt{2} } }$

$\sf{ = \dfrac{ 2 \sqrt{2} \cdot ( 2 + 4)}{ 5\sqrt{2} } }$

$\sf{ = \dfrac{ 2 \sqrt{2} \cdot6}{ 5\sqrt{2} } }$

$\sf{ = \dfrac{ 12 \sqrt{2}}{ 5\sqrt{2} } }$

$\sf{ = \dfrac{ 12}{ 5 } }$