What is the limit as x approaches pi/3 of (1-2cos(x))/(pi-3x) ?

The limit as x approaches pi/3 of (1-2cos(x))/(pi-3x) is -1/(sqrt(3))

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limit as x approaches pi/3 of (1-2cos(x))/(pi-3x)

x \to \frac{π}{3} lim (\frac{1 - 2 cos ( x )}{π - 3 x})

- \frac{1}{3}

Decimal Notation

- 0.57735 \dots

Solution steps

x \to \frac{π}{3} lim (\frac{1 - 2 cos ( x )}{π - 3 x})

Apply L'Hopital's Rule

: x \to \frac{π}{3} lim (\frac{2 sin ( x )}{- 3})

= x \to \frac{π}{3} lim (\frac{2 sin ( x )}{- 3})

Plug in the value

x = \frac{π}{3}

= \frac{2 sin ( \frac{π}{3} )}{- 3}

Simplify

\frac{2 sin ( \frac{π}{3} )}{- 3} : - \frac{1}{3}

= - \frac{1}{3}

x \to - 2 lim (3 x - 2)

x \to 0 lim ((csc (x))^{x})

x \to 0 lim (x^{3} + 3 x^{2} - x + 1)

x \to \infty lim (x - 3 x^{2})

x \to \frac{x}{2} lim (3 sec (x) - 3 tan (x))

What is the limit as x approaches pi/3 of (1-2cos(x))/(pi-3x) ?
The limit as x approaches pi/3 of (1-2cos(x))/(pi-3x) is -1/(sqrt(3))