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

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

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limit as x approaches 0 of xcot(1/4 x)

x \to 0 lim (x cot (\frac{1}{4} x))

4

Solution steps

x \to 0 lim (x cot (\frac{1}{4} x))

Express with sin, cos

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

Use the basic trigonometric identity:

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

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

Rewrite for L'Hopital

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

Apply L'Hopital's Rule

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

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

Plug in the value

x = 0

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

Simplify

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

= 4

x^{^{''}} + 2 x^{^{'}} + 5 x = 0

\frac{e ^{x} + e ^{- x}}{e ^{x} - e ^{- x}}

\int \frac{y}{16 - 9 y ^{4}} d y

\int (2 x^{3} + x) (x^{4} + x^{2}) d x

f (x) = - 2 x + 7

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