\lim_{ x\to\infty } ( \sqrt { x + \sqrt { x } } - \sqrt { x - \sqrt { x } } )

TS ICET 2017 Question Paper with Solutions for online practice

Question : 134 of 200

Marks: +1, -0

\lim ↙{ x→∞ } ( √ { x + √ { x } } - √ { x - √ { x } } )

Solution:

\lim↙{x→∞}(√{x+√{x}}-√{x-√{x}})

=\lim↙{x→∞}(√{x+√{x}}-√{x-√{x}})×\;\;{(√{x+√{x}}+√{x-√{x}})}/{(√{x+√{x}}+√{x-√{x}})}

=\lim↙{x→∞}{(x+√{x})-(x-√{x})}/{√{x+√{x}}+√{x-√{x}}}

=\lim↙{x→∞}{2√{x}}/{√{x+√{x}}+√{x-√{x}}}

=\lim↙{x→∞}{2}/{√{1+\;\;{1}/{√{x}}}+√{1-{1}/{√{x}}}}={2}/{√{1+√{1}}={2}/{2}=1}

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