What is the limit as x approaches-pi of (x+pi)csc(x) ?

The limit as x approaches-pi of (x+pi)csc(x) is -1

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limit as x approaches-pi of (x+pi)csc(x)

x \to - π lim ((x + π) csc (x))

- 1

Solution steps

x \to - π lim ((x + π) csc (x))

Rewrite for L'Hopital

= x \to - π lim (\frac{x + π}{\frac{1}{c s c ( x )}})

Apply L'Hopital's Rule

: x \to - π lim (\frac{1}{cos ( x )})

= x \to - π lim (\frac{1}{cos ( x )})

Plug in the value

x = - π

= \frac{1}{cos ( - π )}

Simplify

\frac{1}{cos ( - π )} : - 1

= - 1

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What is the limit as x approaches-pi of (x+pi)csc(x) ?
The limit as x approaches-pi of (x+pi)csc(x) is -1