What is the limit as x approaches 0 of 6x^2(cot(3x))(csc(2x)) ?

The limit as x approaches 0 of 6x^2(cot(3x))(csc(2x)) is 1

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limit as x approaches 0 of 6x^2(cot(3x))(csc(2x))

x \to 0 lim (6 x^{2} (cot (3 x)) (csc (2 x)))

1

Solution steps

x \to 0 lim (6 x^{2} cot (3 x) csc (2 x))

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

= 6 \cdot x \to 0 lim (x^{2} cot (3 x) csc (2 x))

Express with sin, cos

= 6 \cdot x \to 0 lim (\frac{x ^{2} cos ( 3 x )}{sin ( 3 x ) sin ( 2 x )})

Apply L'Hopital's Rule

: x \to 0 lim (\frac{2 x cos ( 3 x ) - 3 x ^{2} sin ( 3 x )}{cos ( 3 x ) \cdot 3 sin ( 2 x ) + cos ( 2 x ) \cdot 2 sin ( 3 x )})

= 6 \cdot x \to 0 lim (\frac{2 x cos ( 3 x ) - 3 x ^{2} sin ( 3 x )}{cos ( 3 x ) \cdot 3 sin ( 2 x ) + cos ( 2 x ) \cdot 2 sin ( 3 x )})

Apply L'Hopital's Rule

: x \to 0 lim (\frac{2 cos ( 3 x ) - 12 x sin ( 3 x ) - 9 x ^{2} cos ( 3 x )}{- 13 sin ( 3 x ) sin ( 2 x ) + 12 cos ( 3 x ) cos ( 2 x )})

= 6 \cdot x \to 0 lim (\frac{2 cos ( 3 x ) - 12 x sin ( 3 x ) - 9 x ^{2} cos ( 3 x )}{- 13 sin ( 3 x ) sin ( 2 x ) + 12 cos ( 3 x ) cos ( 2 x )})

Plug in the value

x = 0

= 6 \cdot \frac{2 cos ( 3 \cdot 0 ) - 12 \cdot 0 \cdot sin ( 3 \cdot 0 ) - 9 \cdot 0 ^{2} cos ( 3 \cdot 0 )}{- 13 sin ( 3 \cdot 0 ) sin ( 2 \cdot 0 ) + 12 cos ( 3 \cdot 0 ) cos ( 2 \cdot 0 )}

Simplify

6 \cdot \frac{2 cos ( 3 \cdot 0 ) - 12 \cdot 0 \cdot sin ( 3 \cdot 0 ) - 9 \cdot 0 ^{2} cos ( 3 \cdot 0 )}{- 13 sin ( 3 \cdot 0 ) sin ( 2 \cdot 0 ) + 12 cos ( 3 \cdot 0 ) cos ( 2 \cdot 0 )} : 1

= 1

\int_{0}^{1} (x - x^{2})^{2} d x

\int \frac{x - 1}{x ^{2} - 4 x + 3} d x

f (x) = - 2 x^{2}

n = 0 \sum \infty \frac{n}{n + 1} x^{n}

\frac{d}{d x} (5 e^{- 2 x})

What is the limit as x approaches 0 of 6x^2(cot(3x))(csc(2x)) ?
The limit as x approaches 0 of 6x^2(cot(3x))(csc(2x)) is 1