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

The limit as x approaches pi/2-of (tan(x))/(3/(2x-pi)) is -2/3

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

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

- \frac{2}{3}

Decimal

- 0.66666 \dots

Solution steps

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

x \to a lim [c \cdot f (x)] = c \cdot x \to a lim f (x)

= \frac{1}{3} \cdot x \to \frac{π}{2} - lim (\frac{tan ( x )}{\frac{1}{2 x - π}})

Apply L'Hopital's Rule

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

Simplify

\frac{sec ^{2} ( x )}{- \frac{2}{( 2 x - π ) ^{2}}} : - \frac{1}{2} sec^{2} (x) (2 x - π)^{2}

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

x \to a lim [c \cdot f (x)] = c \cdot x \to a lim f (x)

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

Rewrite for L'Hopital

= \frac{1}{3} (- \frac{1}{2} \cdot x \to \frac{π}{2} - lim (\frac{( - π + 2 x ) ^{2}}{\frac{1}{s e c ^{2} ( x )}}))

Apply L'Hopital's Rule

= \frac{1}{3} (- \frac{1}{2} \cdot x \to \frac{π}{2} - lim (\frac{4 ( - π + 2 x )}{- sin ( 2 x )}))

Apply L'Hopital's Rule

= \frac{1}{3} (- \frac{1}{2} \cdot x \to \frac{π}{2} - lim (\frac{8}{- cos ( 2 x ) \cdot 2}))

Plug in the value

x = \frac{π}{2}

= \frac{1}{3} (- \frac{1}{2} \cdot \frac{8}{- cos ( 2 \cdot \frac{π}{2} ) \cdot 2})

Simplify

\frac{1}{3} (- \frac{1}{2} \cdot \frac{8}{- cos ( 2 \cdot \frac{π}{2} ) \cdot 2}) : - \frac{2}{3}

= - \frac{2}{3}

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