What is the limit as x approaches 0 of x/(tan(5x)) ?

The limit as x approaches 0 of x/(tan(5x)) is 1/5

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limit as x approaches 0 of x/(tan(5x))

x \to 0 lim (\frac{x}{tan ( 5 x )})

\frac{1}{5}

Decimal

0.2

Solution steps

x \to 0 lim (\frac{x}{tan ( 5 x )})

Express with sin, cos

= x \to 0 lim (\frac{x cos ( 5 x )}{sin ( 5 x )})

Use the basic trigonometric identity:

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

= x \to 0 lim (x cot (5 x))

Rewrite for L'Hopital

= x \to 0 lim (\frac{x}{\frac{1}{c o t ( 5 x )}})

Apply L'Hopital's Rule

= x \to 0 lim (\frac{1}{sec ^{2} ( 5 x ) \cdot 5})

Plug in the value

x = 0

= \frac{1}{sec ^{2} ( 5 \cdot 0 ) \cdot 5}

sec^{2} (5 \cdot 0) \cdot 5 = 5

= \frac{1}{5}

What is the limit as x approaches 0 of x/(tan(5x)) ?
The limit as x approaches 0 of x/(tan(5x)) is 1/5