Question

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Answer 1

Solution :

$\sf \displaystyle \: \lim_{ \sf \: x \to \: a} \: \sf \dfrac{x \sqrt{x} \: - \: a\sqrt{a} }{x - a} \\ \\ \sf \: if \: we \: put \: x = a \: we \: get \\ \sf \: \implies \: \dfrac{a \sqrt{a} - a \sqrt{a} }{a - a} \: = \: \dfrac{0}{0}$

Therefore applying , L-Hospital rule

$\sf \displaystyle \: \lim_{ \sf \: x \to \: a} \: \sf \dfrac{x \sqrt{x} \: - \: a\sqrt{a} }{x - a} \\ \\ \sf \implies \: \sf \displaystyle \: \lim_{ \sf \: x \to \: a} \: \sf \dfrac{ \dfrac{d}{dx} \bigg(x \sqrt{x} \: - \: a\sqrt{a} \bigg) }{ \dfrac{d}{dx} (x - a)} \\ \\ \sf \implies \: \sf \displaystyle \: \lim_{ \sf \: x \to \: a} \: \sf \dfrac{ \dfrac{d}{dx} \bigg(x \sqrt{x} \: \bigg) - \: \dfrac{d}{dx} \bigg(a\sqrt{a} \bigg) }{ \dfrac{d}{dx} (x) - \dfrac{d}{dx} (a)}$

Note - Here a is constant, and diffrenciation of constant w.r.t x is zero

$\sf \implies \: \sf \displaystyle \: \lim_{ \sf \: x \to \: a} \: \sf \dfrac{ \dfrac{d}{dx} \bigg(x \sqrt{x} \: \bigg) - 0}{ \dfrac{d}{dx} (x) - 0} \\ \\ \sf \implies \: \sf \displaystyle \: \lim_{ \sf \: x \to \: a} \: \sf \dfrac{ \dfrac{d}{dx} \bigg(x \sqrt{x} \: \bigg) }{ \dfrac{d}{dx} (x) }$

Note - Use products rule of diffrenciation in numerator

$\sf \implies \: \sf \displaystyle \: \lim_{ \sf \: x \to \: a} \: \sf \dfrac{ \bigg( \sqrt{x} \times \dfrac{d}{dx} (x) \bigg) \: + \: \bigg(x \: \times \: \dfrac{d}{dx}( \sqrt{x} ) \bigg) }{ \dfrac{d}{dx} (x) }$

Note - identity used

$\sf \bullet \dfrac{d}{dx} {x}^{m} \: = m \times {x}^{(m - 1)}$

$\sf \implies \: \sf \displaystyle \: \lim_{ \sf \: x \to \: a} \: \sf \dfrac{ \bigg( \sqrt{x} \times 1 \bigg) \: + \: \bigg(x \: \times \: \dfrac{1}{2 \sqrt{x} } \bigg) }{ 1 } \\ \\ \sf \implies \: \sf \displaystyle \: \lim_{ \sf \: x \to \: a} \: \bigg( \sf \sqrt{x} \: + \: \: \dfrac{x}{2 \sqrt{x} } \bigg) \\ \\ \sf \implies \: \sf \displaystyle \: \lim_{ \sf \: x \to \: a} \: \bigg( \sf \sqrt{x} \: + \: \: \dfrac{ \sqrt{x} }{2 } \bigg)$

Now in above put limit { that is x = a }

$\sf \implies \: \sqrt{a} + \dfrac{ \sqrt{a} }{2} \\ \\ \sf \implies \: \sqrt{a} \bigg(1 + \dfrac{1}{2} \bigg) \\ \\ \sf \implies \: \sqrt{a} \bigg( \dfrac{3}{2} \bigg) \\ \\ \sf \implies \: \bold{\dfrac{3 \sqrt{a} }{2} }$

ANSWER : (3/2)√a