First slide

Introduction to limits

Question

The value of $\lim_{h \to 0} \{\frac{\sqrt{3} \sin (\frac{π}{6} + h) - \cos (\frac{π}{6} + h)}{\sqrt{3} h (\sqrt{3} \cosh - \sinh)}\}$

Moderate

A

$\frac{2}{\sqrt{3}}$
B

$\frac{4}{3}$
C

$\frac{2}{3}$
D

$\frac{3}{4}$

Solution

$\begin{matrix} \lim_{h \to 0} 2 \{\frac{\sqrt{3} \sin (\frac{π}{6} + h) - \cos (\frac{π}{6} + h)}{\sqrt{3} h (\sqrt{3} \cosh - \sinh)}\} = \lim_{h \to 0} \frac{2}{\sqrt{3} \times \sqrt{3}} \{\frac{\sqrt{3} \sin (\frac{π}{6} + h) - \cos (\frac{π}{6} + h)}{h}\} \\ = \frac{2}{3} \underset{h \to 0}{l i m} \frac{\sqrt{3} \cos (\frac{π}{6} + h) + \sin (\frac{π}{6} + h)}{1} L H o s p i t a l r u l e \\ = \frac{2}{3} \times (\frac{3}{2} + \frac{1}{2}) \\ = \frac{4}{3} \end{matrix}$

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