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 (2\pi )/(3)^+}~csc(x)\implies \lim\limits_{x\to (2\pi )/(3)^+}~\cfrac{1}{sin(x)}\implies \cfrac{1}{sin\left( (2\pi )/(3) \right)}\implies \cfrac{1}{(√(3))/(2)}\implies \cfrac{(1)/(1)}{(√(3))/(2)}`


`\bf -------------------------------\\\\ \cfrac{(a)/(b)}{\frac{c}{{{ d}}}}\implies \cfrac{a}{b}\cdot \cfrac{{{ d}}}{c}\qquad thus\\\\ -------------------------------\\\\ \cfrac{1}{1}\cdot \cfrac{2}{√(3)}\implies \cfrac{1\cdot 2}{1\cdot √(3)}\implies \cfrac{2}{√(3)} \\\\\\ \textit{and now, rationalizing the denominator, we get}\implies \cfrac{2√(3)}{3}`