Root over limit..

Calculus Level 2

Evaluate

lim x x ( x + 20 x ) \lim _{ x\rightarrow \infty }{ \quad \sqrt { x } } \left( \sqrt { x+20 } -\sqrt { x } \right)


The answer is 10.

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1 solution

Karen Vardanyan
Jun 7, 2014

l i m x x ( x + 20 x ) = l i m x x ( x + 20 x ) ( x + 20 + x ) ( x + 20 + x ) = l i m x 20 x ( x + 20 + x ) = l i m x 20 ( 1 + 20 x + 1 ) = 20 2 = 10 { lim }_{ x\rightarrow \infty }\sqrt { x } (\sqrt { x+20 } -\sqrt { x } )={ lim }_{ x\rightarrow \infty }\frac { \sqrt { x } (\sqrt { x+20 } -\sqrt { x } )(\sqrt { x+20 } +\sqrt { x } ) }{ (\sqrt { x+20 } +\sqrt { x } ) } ={ lim }_{ x\rightarrow \infty }\frac { 20\sqrt { x } }{ (\sqrt { x+20 } +\sqrt { x } ) } ={ lim }_{ x\rightarrow \infty }\frac { 20 }{ (\sqrt { 1+\frac { 20 }{ x } } +1) } =\frac { 20 }{ 2 } =10

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