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

The limit as x approaches 0 of (tan(x))/(x^2cot(3x)) is 3

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limit as x approaches 0 of (tan(x))/(x^2cot(3x))

x \to 0 lim (\frac{tan ( x )}{x ^{2} cot ( 3 x )})

3

Solution steps

x \to 0 lim (\frac{tan ( x )}{x ^{2} cot ( 3 x )})

Express with sin, cos

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

Apply L'Hopital's Rule

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

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

Apply L'Hopital's Rule

: x \to 0 lim (\frac{- 10 sin ( x ) sin ( 3 x ) + 6 cos ( x ) cos ( 3 x )}{- 10 x ^{2} cos ( x ) cos ( 3 x ) + 6 x ^{2} sin ( x ) sin ( 3 x ) - 12 x sin ( 3 x ) cos ( x ) - 4 x sin ( x ) cos ( 3 x ) + 2 cos ( 3 x ) cos ( x )})

= x \to 0 lim (\frac{- 10 sin ( x ) sin ( 3 x ) + 6 cos ( x ) cos ( 3 x )}{- 10 x ^{2} cos ( x ) cos ( 3 x ) + 6 x ^{2} sin ( x ) sin ( 3 x ) - 12 x sin ( 3 x ) cos ( x ) - 4 x sin ( x ) cos ( 3 x ) + 2 cos ( 3 x ) cos ( x )})

Plug in the value

x = 0

= \frac{- 10 sin ( 0 ) sin ( 3 \cdot 0 ) + 6 cos ( 0 ) cos ( 3 \cdot 0 )}{- 10 \cdot 0 ^{2} cos ( 0 ) cos ( 3 \cdot 0 ) + 6 \cdot 0 ^{2} sin ( 0 ) sin ( 3 \cdot 0 ) - 12 \cdot 0 \cdot sin ( 3 \cdot 0 ) cos ( 0 ) - 4 \cdot 0 \cdot sin ( 0 ) cos ( 3 \cdot 0 ) + 2 cos ( 3 \cdot 0 ) cos ( 0 )}

Simplify

\frac{- 10 sin ( 0 ) sin ( 3 \cdot 0 ) + 6 cos ( 0 ) cos ( 3 \cdot 0 )}{- 10 \cdot 0 ^{2} cos ( 0 ) cos ( 3 \cdot 0 ) + 6 \cdot 0 ^{2} sin ( 0 ) sin ( 3 \cdot 0 ) - 12 \cdot 0 \cdot sin ( 3 \cdot 0 ) cos ( 0 ) - 4 \cdot 0 \cdot sin ( 0 ) cos ( 3 \cdot 0 ) + 2 cos ( 3 \cdot 0 ) cos ( 0 )} : 3

= 3

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

x \to 0 lim (\frac{x}{( x - 1 ) ^{2}})

x \to 0 lim (arcsin (\frac{1}{x}))

x \to - 3 lim (\frac{2 x}{x - 3})

x \to 3 lim (arcsin (\frac{x}{3}))

What is the limit as x approaches 0 of (tan(x))/(x^2cot(3x)) ?
The limit as x approaches 0 of (tan(x))/(x^2cot(3x)) is 3