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

The limit as x approaches 0 of x/(tan(3x)) is 1/3

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limit as x approaches 0 of x/(tan(3x))

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

\frac{1}{3}

Decimal

0.33333 \dots

Solution steps

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

Express with sin, cos

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

Use the basic trigonometric identity:

\frac{cos ( x )}{sin ( x )} = cot (x)

= x \to 0 lim (x cot (3 x))

Rewrite for L'Hopital

= x \to 0 lim (\frac{x}{\frac{1}{c o t ( 3 x )}})

Apply L'Hopital's Rule

: x \to 0 lim (\frac{1}{sec ^{2} ( 3 x ) \cdot 3})

= x \to 0 lim (\frac{1}{sec ^{2} ( 3 x ) \cdot 3})

Plug in the value

x = 0

= \frac{1}{sec ^{2} ( 3 \cdot 0 ) \cdot 3}

sec^{2} (3 \cdot 0) \cdot 3 = 3

= \frac{1}{3}

\int 8 tan^{3} (x) d x

\frac{\partial}{\partial x} (C e^{2 x} - 3)

n = 0 \sum \infty (x - 2 x^{2})^{n}

x \to 0 + lim (x^{12} \cdot ln (x))

y^{^{''}} + 43 y^{^{'}} + 12 y = 0

What is the limit as x approaches 0 of x/(tan(3x)) ?
The limit as x approaches 0 of x/(tan(3x)) is 1/3