What is the limit as x approaches 0 of x(csc(4x)) ?

The limit as x approaches 0 of x(csc(4x)) is 1/4

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limit as x approaches 0 of x(csc(4x))

x \to 0 lim (x (csc (4 x)))

\frac{1}{4}

Decimal Notation

0.25

Solution steps

x \to 0 lim (x csc (4 x))

Express with sin, cos

= x \to 0 lim (\frac{x}{sin ( 4 x )})

Apply L'Hopital's Rule

: x \to 0 lim (\frac{1}{cos ( 4 x ) \cdot 4})

= x \to 0 lim (\frac{1}{cos ( 4 x ) \cdot 4})

Plug in the value

x = 0

= \frac{1}{cos ( 4 \cdot 0 ) \cdot 4}

cos (4 \cdot 0) \cdot 4 = 4

= \frac{1}{4}

x \to 0 lim (sin^{3} (x))

x \to \infty lim (\frac{cosh ( x )}{e ^{x}})

x \to 0 lim (sin^{2} (\frac{1}{x}))

x \to - 1 lim (3 x^{3} - 4 x^{2} - 3)

x \to 0 lim (e^{\frac{2}{x}})

What is the limit as x approaches 0 of x(csc(4x)) ?
The limit as x approaches 0 of x(csc(4x)) is 1/4