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

The limit as x approaches 0 of x^2*csc^2(x) is 1

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

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

1

Solution steps

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

Rewrite for L'Hopital

= x \to 0 lim (\frac{x ^{2}}{\frac{1}{c s c ^{2} ( x )}})

Apply L'Hopital's Rule

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

Apply L'Hopital's Rule

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

Plug in the value

x = 0

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

Simplify

\frac{2}{cos ( 2 \cdot 0 ) \cdot 2} : 1

= 1

What is the limit as x approaches 0 of x^2*csc^2(x) ?
The limit as x approaches 0 of x^2*csc^2(x) is 1