What is the limit as x approaches 0 of (csc(-6x))/(csc(10x)) ?

The limit as x approaches 0 of (csc(-6x))/(csc(10x)) is -5/3

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limit as x approaches 0 of (csc(-6x))/(csc(10x))

x \to 0 lim (\frac{csc ( - 6 x )}{csc ( 10 x )})

- \frac{5}{3}

Decimal

- 1.66666 \dots

Solution steps

x \to 0 lim (\frac{csc ( - 6 x )}{csc ( 10 x )})

Express with sin, cos

= x \to 0 lim (\frac{sin ( 10 x )}{sin ( - 6 x )})

Apply L'Hopital's Rule

: x \to 0 lim (\frac{cos ( 10 x ) \cdot 10}{- cos ( 6 x ) \cdot 6})

= x \to 0 lim (\frac{cos ( 10 x ) \cdot 10}{- cos ( 6 x ) \cdot 6})

Plug in the value

x = 0

= \frac{cos ( 10 \cdot 0 ) \cdot 10}{- cos ( 6 \cdot 0 ) \cdot 6}

Simplify

\frac{cos ( 10 \cdot 0 ) \cdot 10}{- cos ( 6 \cdot 0 ) \cdot 6} : - \frac{5}{3}

= - \frac{5}{3}

x \to 0 lim (\frac{ln ( 1 + 6 x )}{x})

x \to 1 lim ((x - 1) ln (x))

x \to - 7 lim (\frac{x ^{2} + 16 x + 63}{- 63 - 2 x + x ^{2}})

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

x \to π lim (x - sin (x))

What is the limit as x approaches 0 of (csc(-6x))/(csc(10x)) ?
The limit as x approaches 0 of (csc(-6x))/(csc(10x)) is -5/3