Question

Lim as x approaches 2pi/3 from the right of csc x, solve by substituting csc with sin

Answer 1

$\bf csc(\theta)=\cfrac{1}{sin(\theta)} \\\\ -------------------------------\\\\ \lim\limits_{x\to \frac{2\pi }{3}^+}~csc(x)\implies \lim\limits_{x\to \frac{2\pi }{3}^+}~\cfrac{1}{sin(x)}\implies \cfrac{1}{sin\left( \frac{2\pi }{3} \right)}\implies \cfrac{1}{\frac{\sqrt{3}}{2}}\implies \cfrac{\frac{1}{1}}{\frac{\sqrt{3}}{2}}$

$\bf -------------------------------\\\\ \cfrac{\frac{a}{b}}{\frac{c}{{{ d}}}}\implies \cfrac{a}{b}\cdot \cfrac{{{ d}}}{c}\qquad thus\\\\ -------------------------------\\\\ \cfrac{1}{1}\cdot \cfrac{2}{\sqrt{3}}\implies \cfrac{1\cdot 2}{1\cdot \sqrt{3}}\implies \cfrac{2}{\sqrt{3}} \\\\\\ \textit{and now, rationalizing the denominator, we get}\implies \cfrac{2\sqrt{3}}{3}$