What is the limit as x approaches 0 of xcsc(5x) ?

The limit as x approaches 0 of xcsc(5x) is 1/5

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limit as x approaches 0 of xcsc(5x)

x \to 0 lim (x csc (5 x))

\frac{1}{5}

Decimal

0.2

Solution steps

x \to 0 lim (x csc (5 x))

Express with sin, cos

= x \to 0 lim (\frac{x}{sin ( 5 x )})

Apply L'Hopital's Rule

: x \to 0 lim (\frac{1}{cos ( 5 x ) \cdot 5})

= x \to 0 lim (\frac{1}{cos ( 5 x ) \cdot 5})

Plug in the value

x = 0

= \frac{1}{cos ( 5 \cdot 0 ) \cdot 5}

cos (5 \cdot 0) \cdot 5 = 5

= \frac{1}{5}

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

x \to \infty lim (23)

x \to 0 lim (\frac{3 27 x ^{3} - 2 x + 5}{3 x - 1})

x \to \infty lim (\frac{2}{x} + 3)

x \to \infty lim ((1 + a x)^{\frac{b}{x}})

What is the limit as x approaches 0 of xcsc(5x) ?
The limit as x approaches 0 of xcsc(5x) is 1/5