What is the limit as x approaches 0+of xcot(x)csc(x) ?

The limit as x approaches 0+of xcot(x)csc(x) is infinity

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limit as x approaches 0+of xcot(x)csc(x)

x \to 0 + lim (x cot (x) csc (x))

\infty

Solution steps

x \to 0 + lim (x cot (x) csc (x))

Express with sin, cos

= x \to 0 + lim (\frac{x cos ( x )}{sin ^{2} ( x )})

Apply L'Hopital's Rule

: x \to 0 + lim (\frac{cos ( x ) - x sin ( x )}{sin ( 2 x )})

= x \to 0 + lim (\frac{cos ( x ) - x sin ( x )}{sin ( 2 x )})

Use the basic trigonometric identity:

\frac{1}{sin ( x )} = csc (x)

= x \to 0 + lim (csc (2 x) (cos (x) - x sin (x)))

x \to a lim [f (x) \cdot g (x)] = x \to a lim f (x) \cdot x \to a lim g (x)

With the exception of indeterminate form

= x \to 0 + lim (csc (2 x)) \cdot x \to 0 + lim (cos (x) - x sin (x))

x \to 0 + lim (csc (2 x)) = \infty

x \to 0 + lim (cos (x) - x sin (x)) = 1

= \infty \cdot 1

Apply Infinity Property:

c \cdot \infty = \infty

= \infty

x^{4}, at x = 3

\frac{d}{d x} (\frac{16}{( 1 + 2 x ) ^{3}})

arctan (cos (3 x))

\int \frac{x - 1}{x ^{3}} d x

f (y) = 2 e^{y}

What is the limit as x approaches 0+of xcot(x)csc(x) ?
The limit as x approaches 0+of xcot(x)csc(x) is infinity