What is the limit as x approaches 0 of x^3*csc^2(x) ?

The limit as x approaches 0 of x^3*csc^2(x) is 0

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limit as x approaches 0 of x^3*csc^2(x)

x \to 0 lim (x^{3} \cdot csc^{2} (x))

0

Solution steps

x \to 0 lim (x^{3} csc^{2} (x))

Express with sin, cos

= x \to 0 lim (\frac{x ^{3}}{sin ^{2} ( x )})

Apply L'Hopital's Rule

: x \to 0 lim (\frac{3 x ^{2}}{sin ( 2 x )})

= x \to 0 lim (\frac{3 x ^{2}}{sin ( 2 x )})

Apply L'Hopital's Rule

: x \to 0 lim (\frac{6 x}{cos ( 2 x ) \cdot 2})

= x \to 0 lim (\frac{6 x}{cos ( 2 x ) \cdot 2})

Plug in the value

x = 0

= \frac{6 \cdot 0}{cos ( 2 \cdot 0 ) \cdot 2}

Simplify

\frac{6 \cdot 0}{cos ( 2 \cdot 0 ) \cdot 2} : 0

= 0

f (x) = sin^{5} (2 x) cos^{3} (2 x)

(x^{2} - 1) y^{^{'}} = 2 y

y = \frac{x ^{2} + 1}{3 x + 1}, at x = 0

\frac{\partial}{\partial y} (A x + B y^{2} + C x^{2} y)

y d x + (y - x) d y = 0

What is the limit as x approaches 0 of x^3*csc^2(x) ?
The limit as x approaches 0 of x^3*csc^2(x) is 0