Question

Solve the question...​

Answer 1

Solution

$\bf we \: have$

$\to \tt \: \displaystyle \lim _{ \tt \: x \to4} \tt \bigg( \frac{3 - \sqrt{5 + x} }{1 - \sqrt{5 - x} } \bigg)$

$\bf \: Now \: put \: x = 4 \: on \: given \: equation \: to \: finding \: which \: form$

$\tt \to \: \bigg( \dfrac{3 - \sqrt{5 + 4} }{1 - \sqrt{5 - 4} } \bigg)$

$\tt \to \bigg( \dfrac{3 - \sqrt{9} }{1 - \sqrt{1} } \bigg)$

$\tt \to \: \bigg( \dfrac{3 - 3}{1 - 1} \bigg)$

$\tt \to \dfrac{0}{0}$

$\bf \: Its \: \dfrac{0}{0} \: form \: so \: we \: use \: L'Hospital \: rule$

$\to \tt \: \displaystyle \lim _{ \tt \: x \to0} \tt \bigg( \dfrac{\dfrac{d(3 - \sqrt{5 + x})}{dx} }{ \dfrac{d(1 - \sqrt{5 - x} )}{dx}} \bigg)$

$\tt \to \: \displaystyle \lim _{ \tt \: x \to4} \tt \bigg( \dfrac{ \dfrac{ - 1}{2 \sqrt{5 + x} } }{ \dfrac{ - 1}{2 \sqrt{5 - x} } \times - 1 } \bigg)$

$\tt \to \: \displaystyle \lim _{ \tt \: x \to4} \tt\bigg( \frac{ - 1}{2 \sqrt{5 + x} } \times \frac{2 \sqrt{5 - x} }{1} \bigg)$

$\tt \to \: \displaystyle \lim _{ \tt \: x \to4} \tt\bigg( \frac{ - 2 \sqrt{5 - x} }{2 \sqrt{5 + x} } \bigg)$

$\tt \to \: \displaystyle \lim _{ \tt \: x \to4} \tt\bigg( \frac{ - \sqrt{5 - x} }{ \sqrt{5 + x} } \bigg)$

$\bf \: Now \: put \: the \: value \: x = 4$

$\tt \to \bigg( \dfrac{ - \sqrt{5 - 4} }{ \sqrt{5 + 4} } \bigg)$

$\tt \to \dfrac{ - \sqrt{1} }{ \sqrt{9} }$

$\tt \to \: \dfrac{ - 1}{3}$

$\bf \: Answer$

$\tt \to \: \dfrac{ - 1}{3}$