The value of limh→0 23sin⁡π6+h−cos⁡π6+hh(3sinh−cosh) is

The value of $\underset{h\to 0}{lim} \frac{2\left\{\sqrt{3}\mathrm{sin}\left(\frac{\pi }{6}+h\right)-\mathrm{cos}\left(\frac{\pi }{6}+h\right)\right\}}{h\left(\sqrt{3}\mathrm{sinh}-\mathrm{cosh}\right)}$ is

1. A

$-4$

2. B

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

3. C

4

4. D

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

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Solution:

$\underset{h\to 0}{lim} \frac{2\left\{\sqrt{3}\mathrm{sin}\left(\frac{\pi }{6}+h\right)-\mathrm{cos}\left(\begin{array}{l}\pi \\ 6\end{array}+h\right)\right\}}{h\left(\sqrt{3}\mathrm{sinh}-\mathrm{cosh}\right)}$

$\begin{array}{l}=\underset{h\to 0}{lim} \frac{4\left\{\frac{\sqrt{3}}{2}\mathrm{sin}\left(\frac{\pi }{6}+h\right)-\frac{1}{2}\mathrm{cos}\left(\frac{\pi }{6}+h\right)\right\}}{h\left(\sqrt{3}\mathrm{sinh}-\mathrm{cosh}\right)}\\ =\underset{h\to 0}{lim} \frac{4\left\{\mathrm{sin}\left(\frac{\pi }{6}+h\right)\mathrm{cos}\frac{\pi }{6}-\mathrm{cos}\left(\frac{\pi }{6}+h\right)\mathrm{sin}\frac{\pi }{6}\right\}}{h\left(\sqrt{3}\mathrm{sinh}-\mathrm{cosh}\right)}\\ =\underset{h\to 0}{lim} \frac{4\mathrm{sin}\left(\frac{\pi }{6}+h-\frac{\pi }{6}\right)}{h\left(\sqrt{3}\mathrm{sinh}-\mathrm{cosh}\right)}=4\underset{h\to 0}{lim} \frac{\mathrm{sinh}}{h\left(\sqrt{3}\mathrm{sinh}-\mathrm{cosh}\right)}=-4\end{array}$

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