The value of lim h - 0 2-3sin-6-h-cos-6-h-3 h-3cosh -sin h is -A- 3-42- 2-3C- -3D- 2-31

Lim_h→0 2{√(3)sin ( π6+h ) cos ( π6+h nbsp; )√(3) h(√(3)cos h sin h)} =lim_h→0 42√(3){cos ( π6)sin ( π6+h ) cos ( π6+h nbsp; )sin ( π6) h[cos ( π6)cos h ...

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Standard XII

Mathematics

Standard Limits to Remove Indeterminate Form

The value of ...

Question

The value of $lim h \to 0 2 ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ \frac{\sqrt{3} sin (\frac{π}{6} + h) - cos (\frac{π}{6} + h)}{\sqrt{3} h (\sqrt{3} cos h - sin h)} ⎫ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎭$ is :

\frac{3}{4}

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\frac{2}{\sqrt{3}}

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\frac{4}{3}

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\frac{2}{3}

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Solution

The correct option is C $\frac{4}{3}$
$lim h \to 0 2 ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ \frac{\sqrt{3} sin (\frac{π}{6} + h) - cos (\frac{π}{6} + h)}{\sqrt{3} h (\sqrt{3} cos h - sin h)} ⎫ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎭$
$= lim h \to 0 2 ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ \frac{cos (\frac{π}{6}) sin (\frac{π}{6} + h) - cos (\frac{π}{6} + h) sin (\frac{π}{6})}{2 \sqrt{3} h [cos (\frac{π}{6}) cos h - sin h sin (\frac{π}{6})]} ⎫ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎭$
$= lim h \to 0 2 ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ \frac{2 sin (\frac{π}{6} + h - \frac{π}{6})}{2 \sqrt{3} h (cos (h + \frac{π}{6}))} ⎫ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎭$
$= lim h \to 0 2 ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ \frac{2 sin (h)}{2 \sqrt{3} h cos (\frac{π}{6})} ⎫ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎭$
$= 2 \times \frac{2}{2 \sqrt{3}} \times \frac{2}{\sqrt{3}} = \frac{4}{3}$

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Similar questions

lim h \to 0 2 ⎡ ⎣ \frac{\sqrt{3} sin (\frac{π}{6} + h) - cos (\frac{π}{6} + h)}{\sqrt{3} h (\sqrt{3} cosh - sinh)} ⎤ ⎦ =

lim h \to 0 \frac{2 (\sqrt{3} sin (\frac{π}{6} + h) - cos (\frac{π}{6} + h))}{\sqrt{3} h (\sqrt{3} cos h - sin h)}

is equal to

Q. The value of

lim h \to 0 2 ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ \frac{\sqrt{3} sin (\frac{π}{6} + h) - cos (\frac{π}{6} + h)}{\sqrt{3} h (\sqrt{3} cos h - sin h)} ⎫ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎭

is :

\underset{h \to 0}{\lim 2} \{\frac{\sqrt{3} \sin (π / 6 + h) - \cos (π / 6 + h)}{\sqrt{3} h (\sqrt{3} \cos h - \sin h)}\}

is equal to

(a) 2/3
(b) 4/3
(c)

- 2 \sqrt{3}

(d) −4/3

Q. Let

f (x) = 3 x^{10} - 7 x^{8} + 5 x^{6} - 21 x^{3} + 3 x^{2} - 7

. The value of

l i m h \to 0 \frac{f (1 - h) - f (1)}{h^{3} + 3 h}

is equal to

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Standard Limits to Remove Indeterminate Form

Standard XII Mathematics

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The value of limh→0 2⎧⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪⎩√3sin(π6+h)−cos(π6+h)√3 h(√3cosh−sinh)⎫⎪ ⎪ ⎪ ⎪⎬⎪ ⎪ ⎪ ⎪⎭ is :

The value of $lim h \to 0 2 ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ \frac{\sqrt{3} sin (\frac{π}{6} + h) - cos (\frac{π}{6} + h)}{\sqrt{3} h (\sqrt{3} cos h - sin h)} ⎫ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎭$ is :