Question

please do answer this​

Answer 1

$2) \: \: \: \frac{1}{2 \sqrt{x} } \sf \: \: \: \: is \: the \: answer$

Step-by-step explanation:

$\displaystyle \lim_{h\to \: 0} \bf \: \frac{ \sqrt{x + h} - \sqrt{x} }{h} \\ \\ = \displaystyle \lim_{h\to \: 0} \bf \: \frac{( \sqrt{x + h} - \sqrt{x} )( \sqrt{x + h} + \sqrt{x} ) }{h( \sqrt{x + h} + \sqrt{x} )} \\ \\ = \displaystyle \lim_{h\to \: 0} \bf \: \frac{ {( \sqrt{x + h} })^{2} - {( \sqrt{x} })^{2} }{h( \sqrt{x + h} + \sqrt{x}) } \\ \\ = \displaystyle \lim_{h\to \: 0} \bf \: \frac{x + h - x}{h( \sqrt{x + h} + \sqrt{x}) } \\ \\ = \displaystyle \lim_{h\to \: 0} \bf \: \frac{ \cancel h}{ \cancel{h}( \sqrt{x + h} + \sqrt{x} )} \\ \\ = \displaystyle \lim_{h\to \: 0} \bf \: \frac{1}{ \sqrt{x + h} + \sqrt{x} } \\ \\ = \frac{1}{ \sqrt{x + 0} + \sqrt{x} } \\ \\ = \frac{1}{ \sqrt{x} + \sqrt{x} } \\ \\ = \frac{1}{2 \sqrt{x} }$