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  4. limh →0 (2[√3 sin((π/6)+h)- cos((π/6)+h)]/√3h(√3 cos h-sin h)) equal to

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Q. $ \lim_{h \to0} \frac{2\left[\sqrt3\, sin\left(\frac{\pi}{6}+h\right)-\, cos\left(\frac{\pi}{6}+h\right)\right]}{\sqrt3h(\sqrt3\,cos\,h-sin\,h)}$ equal to

Solution:

$\lim_{h\to0} \frac{2\left[\sqrt{3} \sin\left(\frac{\pi}{6}+h\right) -\cos\left(\frac{\pi}{6}+h\right)\right]}{\sqrt{3} h \left(\sqrt{3} \cos\, h - \sin\, h\right)}$
=$ \lim _{h\to 0} \frac{\left[\sqrt{3} \left(\frac{1}{2} \cos\,h +\frac{\sqrt{3}}{2} \sin\, h\right) -\left(\frac{\sqrt{3}}{2}\cos\, h - \frac{1}{2}\sin\,h \right)\right]}{\sqrt{3} h \left(\sqrt{3} \cos \, h -\sin \, h\right)}$
= $\lim _{h\to 0} \frac{\sqrt{3 } \cos\, h +3 \sin \, h - \sqrt{3} \cos\, h +\sin\, h }{\sqrt{3} h\left[\sqrt{3} \cos\, h - sin \, h\right]}$
= $\lim _{h\to 0} \frac{ 4 \frac{\sin\, h}{h}}{\sqrt{3} \left[\sqrt{3 }\cos \, h - \sin \, h\right] } $
= $\frac{4}{\sqrt{3} \left(\sqrt{3} -0\right) } = \frac{4}{3}$