Question

Evaluate
$\rm \displaystyle \lim_{n\to \sqrt{3}}\ \frac{x^{2}-3}{x^{2}+3\sqrt{3}x-12}$

Answer 1

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Answer 2

Solution :

i ) x² - 3

= x² - ( √3 )²

= ( x + √3 )( x - √3 ) ----( 1 )

[Since ,( a+b )( a-b ) = a² - b² ]

ii ) x² + 3√3x - 12

Splitting the middle term , we get

= x² + 4√3x - √3x - 4√3 × √3

= x( x + 4√3 ) - √3( x + 4√3 )

= ( x + 4√3 )( x - √3 ) ----( 2 )

Now ,

$\rm \displaystyle \lim_{n\to \sqrt{3}}\ \frac{x^{2}-3}{x^{2}+3\sqrt{3}x-12}$

= $\rm \displaystyle \lim_{n\to \sqrt{3}}\ \frac{(x+\sqrt{3})(x-\sqrt{3})}{(x+4\sqrt{3})(x-\sqrt{3})}$

= $\rm \displaystyle \lim_{n\to \sqrt{3}}\ \frac{(x+\sqrt{3})}{(x+4\sqrt{3})}$

= ( √3 + √3 )/( √3 + 4√3 )

= (2√3)/(5√3)

= 2/5

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