What is the limit as x approaches 0+of 5(csc(x)-1/x) ?

The limit as x approaches 0+of 5(csc(x)-1/x) is 0

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limit as x approaches 0+of 5(csc(x)-1/x)

x \to 0 + lim (5 (csc (x) - \frac{1}{x}))

0

Solution steps

x \to 0 + lim (5 (csc (x) - \frac{1}{x}))

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

= 5 \cdot x \to 0 + lim (csc (x) - \frac{1}{x})

Express with sin, cos

= 5 \cdot x \to 0 + lim (\frac{1}{sin ( x )} - \frac{1}{x})

Simplify

\frac{1}{sin ( x )} - \frac{1}{x} : \frac{x - sin ( x )}{x sin ( x )}

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

Apply L'Hopital's Rule

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

Apply L'Hopital's Rule

= 5 \cdot x \to 0 + lim (\frac{sin ( x )}{2 cos ( x ) - x sin ( x )})

Plug in the value

x = 0

= 5 \cdot \frac{sin ( 0 )}{2 cos ( 0 ) - 0 \cdot sin ( 0 )}

Simplify

5 \cdot \frac{sin ( 0 )}{2 cos ( 0 ) - 0 \cdot sin ( 0 )} : 0

= 0

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