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

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

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

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

0

Solution steps

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

Express with sin, cos

= 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 )}

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

Apply L'Hopital's Rule

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

Apply L'Hopital's Rule

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

Plug in the value

x = 0

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

Simplify

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

= 0

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