Question

Evaluate:
$lim_{x \to \infty }\sqrt{x} \: ( \sqrt{x + c} - \sqrt{x} )$
​

Answer 1

$\\ \sf \lim_{x \to \infty } \sqrt{x} ( \sqrt{x + c} - \sqrt{x} )\equiv (\infin -\infin)$

$\\ =\sf \lim _{x \to \infty } \ \sqrt{x( \sqrt{x + c} - \sqrt{x} ) ^{2} }$

Upon Rationalising the Denominator of the Numerator to form a complex fraction

$= \sf \sqrt{ \lim_{x \to \infty } \frac{ {c}^{2} x}{( \sqrt{x + c} + \sqrt{x} ) ^{2} } }$

$\\ = \sf \sqrt{\frac{ {c}^{2} }{ \lim_{x \to \infty }(1 + \frac{ \sqrt{x + c} }{ \sqrt{x} } ) {}^{2} } }$

$=\sf \frac{ c}{(1 + \sqrt{ \lim_{x \to \infty }( 1 + \frac{ c}{x} )} ) }$

$\\ \implies \boxed{ \sf \lim_{x \to \infty } \sqrt{x} ( \sqrt{x + c} - \sqrt{x} ) = \frac{c}{2} }$