What is the limit as x approaches (2pi)/3 of (cot(x)+(sqrt(3))/3)/(x-(2pi)/3) ?

The limit as x approaches (2pi)/3 of (cot(x)+(sqrt(3))/3)/(x-(2pi)/3) is -4/3

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limit as x approaches (2pi)/3 of (cot(x)+(sqrt(3))/3)/(x-(2pi)/3)

x \to \frac{2 π}{3} lim (\frac{cot ( x ) + \frac{3}{3}}{x - \frac{2 π}{3}})

- \frac{4}{3}

Decimal

- 1.33333 \dots

Solution steps

x \to \frac{2 π}{3} lim (\frac{cot ( x ) + \frac{3}{3}}{x - \frac{2 π}{3}})

Apply L'Hopital's Rule

= x \to \frac{2 π}{3} lim (\frac{- csc ^{2} ( x )}{1})

Plug in the value

x = \frac{2 π}{3}

= \frac{- csc ^{2} ( \frac{2 π}{3} )}{1}

Simplify

\frac{- csc ^{2} ( \frac{2 π}{3} )}{1} : - \frac{4}{3}

= - \frac{4}{3}

What is the limit as x approaches (2pi)/3 of (cot(x)+(sqrt(3))/3)/(x-(2pi)/3) ?
The limit as x approaches (2pi)/3 of (cot(x)+(sqrt(3))/3)/(x-(2pi)/3) is -4/3