Lim x - asin-x-sin-2-x-a

Lim_x → asin √(x) sin√(a)/x a =lim_x → asin √(x) sin √(a)/(√(x) √(a))× (√(x)+√(a)) =lim_x → a2 sin(√(x) √(a)/2)cos (√(x)+√(a)/2)/(√(x) √(a))× (√(x)+√(a)) =2 ( l ...

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Byju's Answer

Standard XII

Mathematics

Standard Limits to Remove Indeterminate Form

lim x → asin√...

Question

${lim}_{x \to a} \frac{s i n \sqrt{x} - s i n \sqrt{a}}{x - a}$

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Solution

${lim}_{x \to a} \frac{s i n \sqrt{x} - s i n \sqrt{a}}{x - a}$

$= {lim}_{x \to a} \frac{s i n \sqrt{x} - s i n \sqrt{a}}{(\sqrt{x} - \sqrt{a}) \times (\sqrt{x} + \sqrt{a})}$

$= {lim}_{x \to a} \frac{2 s i n (\frac{\sqrt{x} - \sqrt{a}}{2}) c o s (\frac{\sqrt{x} + \sqrt{a}}{2})}{(\sqrt{x} - \sqrt{a}) \times (\sqrt{x} + \sqrt{a})}$

$= 2 ({lim}_{x \to a} \frac{s i n \frac{\sqrt{x} - \sqrt{a}}{2}}{(\frac{\sqrt{x} - \sqrt{a}}{2})}) \times \frac{1}{2} \frac{{lim}_{x \to a} c o s (\frac{\sqrt{x} + \sqrt{a}}{2})}{{lim}_{x \to a} (\sqrt{x} + \sqrt{a})}$

$= 2 \times 1 \times \frac{1}{2} \times c o s \sqrt{a} \times \frac{1}{2 \sqrt{a}}$

$= \frac{c o s \sqrt{a}}{2 \sqrt{a}}$

Suggest Corrections

limx→asin√x−sin√ax−a

limx→asin√x−sin√ax−a =limx→asin√x−sin√a(√x−√a)×(√x+√a) =limx→a2 sin(√x−√a2)cos(√x+√a2)(√x−√a)×(√x+√a) =2(limx→asin√x−√a2(√x−√a2))×12limx→acos(√x+√a2)limx→a(√x+√a) =2×1×12×cos√a×12√a =cos√a2√a

${lim}_{x \to a} \frac{s i n \sqrt{x} - s i n \sqrt{a}}{x - a}$