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

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

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

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

2

Solution steps

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

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

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

Apply exponent rule:

a^{n} \cdot b^{n} = (a \cdot b)^{n}

x^{2} cot^{2} (x) = (x cot (x))^{2}

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

x \to a lim [f (x)]^{b} = [x \to a lim f (x)]^{b}

With the exception of indeterminate form

= 2 (x \to 0 lim (x cot (x)))^{2}

Rewrite for L'Hopital

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

Apply L'Hopital's Rule

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

Plug in the value

x = 0

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

Simplify

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

= 2

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