First slide

Introduction to limits

Question

The value of
$lim_{h \to 0} \frac{2 \{\sqrt{3} \sin (\frac{π}{6} + h) - \cos (\frac{π}{6} + h)\}}{h (\sqrt{3} \sinh - \cosh)}$ is

Moderate

A

$- 4$
B

$- \frac{4}{\sqrt{3}}$
C

4
D

$\frac{4}{\sqrt{3}}$

Solution

$lim_{h \to 0} \frac{2 \{\sqrt{3} \sin (\frac{π}{6} + h) - \cos (\begin{array}{l} π \\ 6 \end{array} + h)\}}{h (\sqrt{3} \sinh - \cosh)}$
$\begin{array}{l} = lim_{h \to 0} \frac{4 \{\frac{\sqrt{3}}{2} \sin (\frac{π}{6} + h) - \frac{1}{2} \cos (\frac{π}{6} + h)\}}{h (\sqrt{3} \sinh - \cosh)} \\ = lim_{h \to 0} \frac{4 \{\sin (\frac{π}{6} + h) \cos \frac{π}{6} - \cos (\frac{π}{6} + h) \sin \frac{π}{6}\}}{h (\sqrt{3} \sinh - \cosh)} \\ = lim_{h \to 0} \frac{4 \sin (\frac{π}{6} + h - \frac{π}{6})}{h (\sqrt{3} \sinh - \cosh)} = 4 lim_{h \to 0} \frac{\sinh}{h (\sqrt{3} \sinh - \cosh)} = - 4 \end{array}$

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