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

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

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

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

\frac{1}{3}

Decimal

0.33333 \dots

Solution steps

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

Express with sin, cos

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

Use the basic trigonometric identity:

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

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

Rewrite for L'Hopital

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

Apply L'Hopital's Rule

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

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

Plug in the value

t = 0

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

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

= \frac{1}{3}

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