What is the limit as x approaches 0 of x/(tan(4x)) ?

The limit as x approaches 0 of x/(tan(4x)) is 1/4

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limit as x approaches 0 of x/(tan(4x))

x \to 0 lim (\frac{x}{tan ( 4 x )})

\frac{1}{4}

Decimal

0.25

Solution steps

x \to 0 lim (\frac{x}{tan ( 4 x )})

Express with sin, cos

= x \to 0 lim (\frac{x cos ( 4 x )}{sin ( 4 x )})

Use the basic trigonometric identity:

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

= x \to 0 lim (x cot (4 x))

Rewrite for L'Hopital

= x \to 0 lim (\frac{x}{\frac{1}{c o t ( 4 x )}})

Apply L'Hopital's Rule

= x \to 0 lim (\frac{1}{sec ^{2} ( 4 x ) \cdot 4})

Plug in the value

x = 0

= \frac{1}{sec ^{2} ( 4 \cdot 0 ) \cdot 4}

sec^{2} (4 \cdot 0) \cdot 4 = 4

= \frac{1}{4}

What is the limit as x approaches 0 of x/(tan(4x)) ?
The limit as x approaches 0 of x/(tan(4x)) is 1/4