What is the limit as x approaches 0 of 3xcsc(5x) ?

The limit as x approaches 0 of 3xcsc(5x) is 3/5

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limit as x approaches 0 of 3xcsc(5x)

x \to 0 lim (3 x csc (5 x))

\frac{3}{5}

Decimal

0.6

Solution steps

x \to 0 lim (3 x csc (5 x))

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

= 3 \cdot x \to 0 lim (x csc (5 x))

Express with sin, cos

= 3 \cdot x \to 0 lim (\frac{x}{sin ( 5 x )})

Apply L'Hopital's Rule

= 3 \cdot x \to 0 lim (\frac{1}{cos ( 5 x ) \cdot 5})

Plug in the value

x = 0

= 3 \cdot \frac{1}{cos ( 5 \cdot 0 ) \cdot 5}

Simplify

3 \cdot \frac{1}{cos ( 5 \cdot 0 ) \cdot 5} : \frac{3}{5}

= \frac{3}{5}

What is the limit as x approaches 0 of 3xcsc(5x) ?
The limit as x approaches 0 of 3xcsc(5x) is 3/5