What is the limit as x approaches 0 of xcsc(pix) ?

The limit as x approaches 0 of xcsc(pix) is 1/pi

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limit as x approaches 0 of xcsc(pix)

x \to 0 lim (x csc (π x))

\frac{1}{π}

Decimal Notation

0.31830 \dots

Solution steps

x \to 0 lim (x csc (π x))

Express with sin, cos

= x \to 0 lim (\frac{x}{sin ( π x )})

Apply L'Hopital's Rule

: x \to 0 lim (\frac{1}{cos ( π x ) π})

= x \to 0 lim (\frac{1}{cos ( π x ) π})

Plug in the value

x = 0

= \frac{1}{cos ( π 0 ) π}

cos (π 0) π = π

= \frac{1}{π}

x \to \frac{1}{3} lim (x^{4})

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

(x, y) \to (0, 0) lim (1)

x \to 3 - lim (- x^{2} + 6 x - 5)

y = x^{2} - 3 x, y = 3 x + 7

What is the limit as x approaches 0 of xcsc(pix) ?
The limit as x approaches 0 of xcsc(pix) is 1/pi