Question

Evaluate:
$\rm \displaystyle \lim_{x \to \infty}\ \sqrt{x^{2}+3x}-x$

Answer 1

Solution :

i ) (√x²+3x) - x

={[(√x²+3x)-x][(√x²+3x)+x]}/[(√(x²+3x) + x ]

= [ (√x²+3x)² - x² ]/[(√x²+3x) + x ]

= ( x² + 3x - x² )/{x[√1+(3/x)+ 1 ]}

= 3x/[ x( √(1+3/x) +1 ]

= 3/[√(1 + 3/x) + 1 ] ----( 1 )

Now ,

$\rm \displaystyle \lim_{x \to \infty}\ \sqrt{x^{2}+3x}-x$

= $\rm \displaystyle \lim_{x \to \infty}\ \frac{3}{\sqrt{1+\frac{3}{x}}+1]}$

= 3/[ √(1+0) + 1 ]

= 3/2

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