Question

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

Answer 1

Answer:

$refer \: to \: attachment \\ i \: try \: this$

Answer 2

$\bf \to \: \green{{ \underline{answer}}} \\ \\ \sf \to \: \lim_{x \: \to \: \: \infty } \: \sqrt{x} ( \sqrt{x +c} - \sqrt{x} ) = \dfrac{c}{2}$

$\sf \to \: \orange{{ \underline{step - \: by - \: step - \: explanation}}}$

$\sf \to \: \lim_{ x \: \to \: \infty } = \sqrt{x}( \sqrt{x + c} - \sqrt{x} \\ \\ \sf \to \: \lim_{x \: \to \: \infty } \: = \sqrt{ \infty } ( \sqrt{ \infty \: + c} - \sqrt{ \infty } ) \\ \\ \sf \to \: it \: is \: in \: the \: form \: of \: (\infty \: - \: \infty )\: form \\ \\ \sf \to \: when \: root \: is \: found \: in \: equation \: we \: can \: rationalise \: it \\ \\ \sf \to \: \lim_{x \: \to \: \infty } \: = \frac{ \sqrt{x} ( \sqrt{x + c} - \sqrt{x})( \sqrt{x + c} + \sqrt{x} ) }{ \sqrt{x + c} + \sqrt{x} } \\ \\ \sf \to \: using \: identity \: = {x}^{2} - {y}^{2} = (x + y)(x - y) \\ \\ \sf \to \: \lim_{x \: \to \: \infty } \: = \frac{ \sqrt{x} (x + c - x)}{ \sqrt{x + c} + \sqrt{x} }$

$\sf \to \: \lim_{x \: \to \: \infty } \: \dfrac{c \sqrt{x} }{ \sqrt{x + c} + \sqrt{x} } \\ \\ \sf \to \: put \: x \: = \infty \: in \: equation \\ \\ \sf \to \: \lim_{x \: \to \: \infty } \: = \frac{ c \sqrt{ \infty } }{ \sqrt{ \infty + c} + \sqrt{ \infty } } \\ \\ \sf \to \: it \: is \: in \: the \: form \: of \: \frac{ \infty }{ \infty } form \\ \\ \sf \to \: divide \: numerator \: and \: denominator \: by \: \sqrt{x} \\ \\ \sf \to \: \lim_{x \: \to \: \infty } \: = \frac{c \dfrac{ \sqrt{x} }{ \sqrt{x} } }{ \sqrt{ \dfrac{x}{ x } + \dfrac{c}{ x } } + \frac{ \sqrt{x} }{ \sqrt{x} } } \\ \\ \sf \to \: \lim_{x \: \to \: \infty } \: \dfrac{c}{ \sqrt{1 + \dfrac{c}{x} } + 1} \\ \\ \sf \to \: \lim_{x \: \to \: \infty } \: = \dfrac{c}{2} = answer$