Lim x -2-2-1-sin x-cos 2 x

Lim_x →π/2√(2) √(1+sin x)/cos^2 x =lim_x →π/2√(2) √(1+sin x)/cos^2 x×√(2)+√(1+sin x)/√(2)+√(1+sin x)=lim_x →π/22 1 sin x/cos^2 x(√(2)+√(1)+sin x) =lim_x →π/21 s ...

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Standard XII

Mathematics

Rationalization Method to Remove Indeterminate Form

lim x →π/2√2-...

Question

${lim}_{x \to \frac{π}{2}} \frac{\sqrt{2} - \sqrt{1 + s i n x}}{c o s^{2} x}$

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Solution

${lim}_{x \to \frac{π}{2}} \frac{\sqrt{2} - \sqrt{1 + s i n x}}{c o s^{2} x}$

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

$= {lim}_{x \to \frac{π}{2}} \frac{1 - s i n x}{(1 - s i n^{2} x) (\sqrt{2} + \sqrt{1 + s i n x})} = {lim}_{x \to \frac{π}{2}} \frac{1}{(1 + s i n x) (\sqrt{2} + \sqrt{1 + s i n x})}$

$= \frac{1}{(1 + 1) (\sqrt{2} + \sqrt{2})} = \frac{1}{(4 \sqrt{2})}$

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Similar questions

Q. Evaluate

lim x \to \frac{π}{2} \frac{\sqrt{2} - \sqrt{1 + sin x}}{{cos}^{2} x}

Q. Assertion :

lim x \to 0 \frac{\sqrt{1 - cos 2 x}}{x}

does not exist. Reason:

| sin x | = ⎧ ⎪ ⎨ ⎪ ⎩ \begin{matrix} sin x; & 0 < x < \frac{π}{2} - sin x; & - \frac{π}{2} < x < 0 \end{matrix}

${lim}_{x \to \frac{π}{2}} \frac{1 - s i n x}{c o s^{2} x}$

Find the value of $lim x \to \frac{π}{4} \frac{c o s x - s i n x}{c o s 2 x}$

${lim}_{x \to \frac{π}{2}} \frac{\sqrt{2 - s i n x} - 1}{(\frac{π}{2} - x)^{2}}$

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limx→π2√2−√1+sin xcos2 x

limx→π2√2−√1+sin xcos2 x =limx→π2√2−√1+sin xcos2 x×√2+√1+sin x√2+√1+sin x=limx→π22−1−sin xcos2x(√2+√1+sin x) =limx→π21−sin x(1−sin2x)(√2+√1+sin x)=limx→π21(1+sin x)(√2+√1+sin x) =1(1+1)(√2+√2)=1(4√2)

${lim}_{x \to \frac{π}{2}} \frac{\sqrt{2} - \sqrt{1 + s i n x}}{c o s^{2} x}$