What is the limit as x approaches 0 of x/(tan(7x)) ?

The limit as x approaches 0 of x/(tan(7x)) is 1/7

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limit as x approaches 0 of x/(tan(7x))

x \to 0 lim (\frac{x}{tan ( 7 x )})

\frac{1}{7}

Decimal Notation

0.14285 \dots

Solution steps

x \to 0 lim (\frac{x}{tan ( 7 x )})

Express with sin, cos

= x \to 0 lim (\frac{x cos ( 7 x )}{sin ( 7 x )})

Use the basic trigonometric identity:

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

= x \to 0 lim (x cot (7 x))

Rewrite for L'Hopital

= x \to 0 lim (\frac{x}{\frac{1}{c o t ( 7 x )}})

Apply L'Hopital's Rule

: x \to 0 lim (\frac{1}{sec ^{2} ( 7 x ) \cdot 7})

= x \to 0 lim (\frac{1}{sec ^{2} ( 7 x ) \cdot 7})

Plug in the value

x = 0

= \frac{1}{sec ^{2} ( 7 \cdot 0 ) \cdot 7}

sec^{2} (7 \cdot 0) \cdot 7 = 7

= \frac{1}{7}

\frac{\partial}{\partial x} (e^{- x yz})

\frac{\partial}{\partial x} (ln (x + 3))

3 (1 + x) \frac{d y}{d x} = 2 x y (y^{3} + 1)

f (x) = \frac{8}{x} - \frac{4}{x}

\int cos^{3} (\frac{x}{6}) d x

What is the limit as x approaches 0 of x/(tan(7x)) ?
The limit as x approaches 0 of x/(tan(7x)) is 1/7