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

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

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

x \to 0 - lim (4 (csc (x) - \frac{1}{x}))

0

Solution steps

x \to 0 - lim (4 (csc (x) - \frac{1}{x}))

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

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

Express with sin, cos

= 4 \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 )}

= 4 \cdot 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 )})

= 4 \cdot 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 )})

= 4 \cdot x \to 0 - lim (\frac{sin ( x )}{2 cos ( x ) - x sin ( x )})

Plug in the value

x = 0

= 4 \cdot \frac{sin ( 0 )}{2 cos ( 0 ) - 0 \cdot sin ( 0 )}

Simplify

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

= 0

x \to 7 lim (\frac{x - 7}{x ^{2} - 7})

x \to 0 lim (x^{2} (ln (x^{2})))

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

x \to 3 lim (x^{\frac{1}{x}})

x \to 2 lim (81)

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