hryggcx

2022-06-29

Calculating $\underset{x\to \frac{\pi }{6}}{lim}\left(\frac{\mathrm{sin}\left(x-\frac{\pi }{6}\right)}{\frac{\sqrt{3}}{2}-\mathrm{cos}x}\right)$, without using L'Hospital rule

Perman7z

Expert

Let put $t=x-\frac{\pi }{6}$
we will compute
$\underset{t\to 0}{lim}\frac{\mathrm{sin}\left(t\right)}{\frac{\sqrt{3}}{2}-\mathrm{cos}\left(t+\frac{\pi }{6}\right)}=$
$\underset{t\to 0}{lim}\frac{\mathrm{sin}\left(t\right)}{\frac{\sqrt{3}}{2}\left(1-\mathrm{cos}\left(t\right)\right)+\frac{\mathrm{sin}\left(t\right)}{2}}=$
$\underset{t\to 0}{lim}\frac{1}{\frac{1}{2}+\frac{\sqrt{3}}{2}\frac{{t}^{2}\left(1-\mathrm{cos}\left(t\right)\right)}{{t}^{2}\mathrm{sin}\left(t\right)}}=2.$

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