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Definite Integral as Limit of Sum

limx→ a [ a1...

Question

$lim x \to a {⎧ ⎨ ⎩ ⎡ ⎣ {(\frac{a^{1 / 2} + x^{1 / 2}}{a^{1 / 4} - x^{1 / 4}})}^{- 1} - \frac{2 (a x)^{1 / 4}}{x^{3 / 4} a^{1 / 4} x^{1 / 2} + a^{1 / 2} x^{1 / 4}} - a^{3 / 4} - \sqrt{2^{l o g_{4} a}} ⎤ ⎦ ⎫ ⎬ ⎭}^{8}$ is

A

a

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B

a^{3 / 4}

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C

a^{2}

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D

none of these

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Solution

The correct option is B $a^{2}$
Simplifying the expression in brackets by setting $a^{1 / 4} = b$ and $x^{1 / 4} = y,$ the function whose limit is required can be written as
${⎧ ⎪ ⎨ ⎪ ⎩ {⎡ ⎣ {(\frac{b^{2} + y^{2}}{b - y})}^{- 1} - \frac{2 b y}{y^{3} - b y^{2} + b^{2} y - b^{3}} ⎤ ⎦}^{- 1} - 2^{\frac{1}{2} l o g_{4} b^{4}} ⎫ ⎪ ⎬ ⎪ ⎭}^{8}$
$= {⎧ ⎨ ⎩ {[\frac{b - y}{b^{2} + y^{2}} - \frac{2 b y}{(y - b) (b^{2} + y^{2})}]}^{- 1} - b ⎫ ⎬ ⎭}^{8}$
$= {[\frac{1}{b - y} - b]}^{8} = y^{8} = x^{2}$
Hence the required lim as $x \to a$ is $a^{2}$

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