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$stack L t with x rightwards arrow 2 below open parentheses fraction numerator x to the power of 3 end exponent minus 4 x over denominator x to the power of 3 end exponent minus 8 end fraction close parentheses to the power of negative 1 end exponent minus open parentheses fraction numerator x plus square root of 2 x end root over denominator x minus 2 end fraction minus fraction numerator square root of 2 over denominator square root of x minus square root of 2 end fraction close parentheses to the power of negative 1 end exponent$ is

$\frac{1}{2}$
1
2
none of these

The correct answer is: $fraction numerator 1 over denominator 2 end fraction$

Reqd. limit
$equals stack text Lt end text with x rightwards arrow 2 below fraction numerator left parenthesis x minus 2 right parenthesis left parenthesis x to the power of 2 end exponent plus 2 x plus 4 right parenthesis over denominator x left parenthesis x minus 2 right parenthesis left parenthesis x plus 2 right parenthesis end fraction minus open parentheses fraction numerator square root of x left parenthesis square root of x plus square root of 2 right parenthesis over denominator x minus 2 end fraction minus fraction numerator square root of 2 left parenthesis x minus 2 right parenthesis over denominator square root of x minus square root of 2 end fraction close parentheses to the power of negative 1 end exponent$
$equals stack text Lt end text with x rightwards arrow 2 below open square brackets fraction numerator x to the power of 2 end exponent plus 2 x plus 4 over denominator x left parenthesis x plus 2 right parenthesis end fraction. open parentheses fraction numerator square root of x left parenthesis x minus 2 right parenthesis minus square root of 2 left parenthesis x minus 2 right parenthesis over denominator left parenthesis x minus 2 right parenthesis left parenthesis square root of x minus square root of 2 right parenthesis end fraction close parentheses to the power of negative 1 end exponent close square brackets$
$equals stack text Lt end text with x rightwards arrow 2 below fraction numerator x to the power of 2 end exponent plus 2 x plus 4 over denominator x left parenthesis x plus 2 right parenthesis end fraction minus stack text Lt end text with x rightwards arrow 2 below open parentheses fraction numerator open parentheses square root of x minus square root of 2 close parentheses left parenthesis x minus 2 right parenthesis over denominator left parenthesis x minus 2 right parenthesis left parenthesis square root of x minus square root of 2 right parenthesis end fraction close parentheses to the power of negative 1 end exponent$
$equals fraction numerator 4 plus 4 plus 4 over denominator 2 left parenthesis 2 plus 2 right parenthesis end fraction minus left parenthesis 1 right parenthesis to the power of negative 1 end exponent equals fraction numerator 12 over denominator 8 end fraction minus 1 equals fraction numerator 3 over denominator 2 end fraction minus 1 equals fraction numerator 1 over denominator 2 end fraction$

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