Calculus Examples

$lim_{x \to 0} (x^{2}) \cot (x^{2})$

Step 1

Rewrite $(x^{2}) \cot (x^{2})$ as $\frac{x^{2}}{\frac{1}{\cot (x^{2})}}$ .

$lim_{x \to 0} \frac{x^{2}}{\frac{1}{\cot (x^{2})}}$

Step 2

Set up the limit as a left-sided limit.

$lim_{x \to 0^{-}} \frac{x^{2}}{\frac{1}{\cot (x^{2})}}$

Step 3

Evaluate the left-sided limit.

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Step 3.1

Apply L'Hospital's rule.

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Step 3.1.1

Evaluate the limit of the numerator and the limit of the denominator.

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Step 3.1.1.1

Take the limit of the numerator and the limit of the denominator.

$\frac{lim_{x \to 0^{-}} x^{2}}{lim_{x \to 0^{-}} \frac{1}{\cot (x^{2})}}$

Step 3.1.1.2

Evaluate the limit of the numerator.

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Step 3.1.1.2.1

Move the exponent $2$ from $x^{2}$ outside the limit using the Limits Power Rule.

$\frac{{(lim_{x \to 0^{-}} x)}^{2}}{lim_{x \to 0^{-}} \frac{1}{\cot (x^{2})}}$

Step 3.1.1.2.2

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$\frac{0^{2}}{lim_{x \to 0^{-}} \frac{1}{\cot (x^{2})}}$

Step 3.1.1.2.3

Raising $0$ to any positive power yields $0$ .

$\frac{0}{lim_{x \to 0^{-}} \frac{1}{\cot (x^{2})}}$

Step 3.1.1.3

Evaluate the limit of the denominator.

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Step 3.1.1.3.1

Apply trigonometric identities.

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Step 3.1.1.3.1.1

Rewrite $\cot (x^{2})$ in terms of sines and cosines.

$\frac{0}{lim_{x \to 0^{-}} \frac{1}{\frac{\cos (x^{2})}{\sin (x^{2})}}}$

Step 3.1.1.3.1.2

Multiply by the reciprocal of the fraction to divide by $\frac{\cos (x^{2})}{\sin (x^{2})}$ .

$\frac{0}{lim_{x \to 0^{-}} \frac{\sin (x^{2})}{\cos (x^{2})}}$

Step 3.1.1.3.1.3

Convert from $\frac{\sin (x^{2})}{\cos (x^{2})}$ to $\tan (x^{2})$ .

$\frac{0}{lim_{x \to 0^{-}} \tan (x^{2})}$

Step 3.1.1.3.2

Evaluate the limit.

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Step 3.1.1.3.2.1

Move the limit inside the trig function because tangent is continuous.

$\frac{0}{\tan (lim_{x \to 0^{-}} x^{2})}$

Step 3.1.1.3.2.2

Move the exponent $2$ from $x^{2}$ outside the limit using the Limits Power Rule.

$\frac{0}{\tan ({(lim_{x \to 0^{-}} x)}^{2})}$

Step 3.1.1.3.3

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$\frac{0}{\tan (0^{2})}$

Step 3.1.1.3.4

Simplify the answer.

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Step 3.1.1.3.4.1

Raising $0$ to any positive power yields $0$ .

$\frac{0}{\tan (0)}$

Step 3.1.1.3.4.2

The exact value of $\tan (0)$ is $0$ .

$\frac{0}{0}$

Step 3.1.1.3.4.3

The expression contains a division by $0$ . The expression is undefined.

Undefined

$\frac{0}{0}$

Step 3.1.1.3.5

The expression contains a division by $0$ . The expression is undefined.

Undefined

$\frac{0}{0}$

Step 3.1.1.4

The expression contains a division by $0$ . The expression is undefined.

Undefined

$\frac{0}{0}$

Step 3.1.2

Since $\frac{0}{0}$ is of indeterminate form, apply L'Hospital's Rule. L'Hospital's Rule states that the limit of a quotient of functions is equal to the limit of the quotient of their derivatives.

$lim_{x \to 0^{-}} \frac{x^{2}}{\frac{1}{\cot (x^{2})}} = lim_{x \to 0^{-}} \frac{\frac{d}{d x} [x^{2}]}{\frac{d}{d x} [\frac{1}{\cot (x^{2})}]}$

Step 3.1.3

Find the derivative of the numerator and denominator.

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Step 3.1.3.1

Differentiate the numerator and denominator.

$lim_{x \to 0^{-}} \frac{\frac{d}{d x} [x^{2}]}{\frac{d}{d x} [\frac{1}{\cot (x^{2})}]}$

Step 3.1.3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$lim_{x \to 0^{-}} \frac{2 x}{\frac{d}{d x} [\frac{1}{\cot (x^{2})}]}$

Step 3.1.3.3

Rewrite $\cot (x^{2})$ in terms of sines and cosines.

$lim_{x \to 0^{-}} \frac{2 x}{\frac{d}{d x} [\frac{1}{\frac{\cos (x^{2})}{\sin (x^{2})}}]}$

Step 3.1.3.4

Multiply by the reciprocal of the fraction to divide by $\frac{\cos (x^{2})}{\sin (x^{2})}$ .

$lim_{x \to 0^{-}} \frac{2 x}{\frac{d}{d x} [1 \frac{\sin (x^{2})}{\cos (x^{2})}]}$

Step 3.1.3.5

Write $1$ as a fraction with denominator $1$ .

$lim_{x \to 0^{-}} \frac{2 x}{\frac{d}{d x} [\frac{1}{1} \cdot \frac{\sin (x^{2})}{\cos (x^{2})}]}$

Step 3.1.3.6

Simplify.

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Step 3.1.3.6.1

Rewrite the expression.

$lim_{x \to 0^{-}} \frac{2 x}{\frac{d}{d x} [1 \frac{\sin (x^{2})}{\cos (x^{2})}]}$

Step 3.1.3.6.2

Multiply $\frac{\sin (x^{2})}{\cos (x^{2})}$ by $1$ .

$lim_{x \to 0^{-}} \frac{2 x}{\frac{d}{d x} [\frac{\sin (x^{2})}{\cos (x^{2})}]}$

Step 3.1.3.7

Differentiate using the Quotient Rule which states that $\frac{d}{d x} [\frac{f (x)}{g (x)}]$ is $\frac{g (x) \frac{d}{d x} [f (x)] - f (x) \frac{d}{d x} [g (x)]}{{g (x)}^{2}}$ where $f (x) = \sin (x^{2})$ and $g (x) = \cos (x^{2})$ .

$lim_{x \to 0^{-}} \frac{2 x}{\frac{\cos (x^{2}) \frac{d}{d x} [\sin (x^{2})] - \sin (x^{2}) \frac{d}{d x} [\cos (x^{2})]}{\cos^{2} (x^{2})}}$

Step 3.1.3.8

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \sin (x)$ and $g (x) = x^{2}$ .

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Step 3.1.3.8.1

To apply the Chain Rule, set $u_{1}$ as $x^{2}$ .

$lim_{x \to 0^{-}} \frac{2 x}{\frac{\cos (x^{2}) (\frac{d}{d u_{1}} [\sin (u_{1})] \frac{d}{d x} [x^{2}]) - \sin (x^{2}) \frac{d}{d x} [\cos (x^{2})]}{\cos^{2} (x^{2})}}$

Step 3.1.3.8.2

The derivative of $\sin (u_{1})$ with respect to $u_{1}$ is $\cos (u_{1})$ .

$lim_{x \to 0^{-}} \frac{2 x}{\frac{\cos (x^{2}) (\cos (u_{1}) \frac{d}{d x} [x^{2}]) - \sin (x^{2}) \frac{d}{d x} [\cos (x^{2})]}{\cos^{2} (x^{2})}}$

Step 3.1.3.8.3

Replace all occurrences of $u_{1}$ with $x^{2}$ .

$lim_{x \to 0^{-}} \frac{2 x}{\frac{\cos (x^{2}) (\cos (x^{2}) \frac{d}{d x} [x^{2}]) - \sin (x^{2}) \frac{d}{d x} [\cos (x^{2})]}{\cos^{2} (x^{2})}}$

Step 3.1.3.9

Raise $\cos (x^{2})$ to the power of $1$ .

$lim_{x \to 0^{-}} \frac{2 x}{\frac{\cos^{1} (x^{2}) \cos (x^{2}) \frac{d}{d x} [x^{2}] - \sin (x^{2}) \frac{d}{d x} [\cos (x^{2})]}{\cos^{2} (x^{2})}}$

Step 3.1.3.10

Raise $\cos (x^{2})$ to the power of $1$ .

$lim_{x \to 0^{-}} \frac{2 x}{\frac{\cos^{1} (x^{2}) \cos^{1} (x^{2}) \frac{d}{d x} [x^{2}] - \sin (x^{2}) \frac{d}{d x} [\cos (x^{2})]}{\cos^{2} (x^{2})}}$

Step 3.1.3.11

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$lim_{x \to 0^{-}} \frac{2 x}{\frac{{\cos (x^{2})}^{1 + 1} \frac{d}{d x} [x^{2}] - \sin (x^{2}) \frac{d}{d x} [\cos (x^{2})]}{\cos^{2} (x^{2})}}$

Step 3.1.3.12

Add $1$ and $1$ .

$lim_{x \to 0^{-}} \frac{2 x}{\frac{\cos^{2} (x^{2}) \frac{d}{d x} [x^{2}] - \sin (x^{2}) \frac{d}{d x} [\cos (x^{2})]}{\cos^{2} (x^{2})}}$

Step 3.1.3.13

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$lim_{x \to 0^{-}} \frac{2 x}{\frac{\cos^{2} (x^{2}) (2 x) - \sin (x^{2}) \frac{d}{d x} [\cos (x^{2})]}{\cos^{2} (x^{2})}}$

Step 3.1.3.14

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \cos (x)$ and $g (x) = x^{2}$ .

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Step 3.1.3.14.1

To apply the Chain Rule, set $u_{2}$ as $x^{2}$ .

$lim_{x \to 0^{-}} \frac{2 x}{\frac{\cos^{2} (x^{2}) (2 x) - \sin (x^{2}) (\frac{d}{d u_{2}} [\cos (u_{2})] \frac{d}{d x} [x^{2}])}{\cos^{2} (x^{2})}}$

Step 3.1.3.14.2

The derivative of $\cos (u_{2})$ with respect to $u_{2}$ is $- \sin (u_{2})$ .

$lim_{x \to 0^{-}} \frac{2 x}{\frac{\cos^{2} (x^{2}) (2 x) - \sin (x^{2}) (- \sin (u_{2}) \frac{d}{d x} [x^{2}])}{\cos^{2} (x^{2})}}$

Step 3.1.3.14.3

Replace all occurrences of $u_{2}$ with $x^{2}$ .

$lim_{x \to 0^{-}} \frac{2 x}{\frac{\cos^{2} (x^{2}) (2 x) - \sin (x^{2}) (- \sin (x^{2}) \frac{d}{d x} [x^{2}])}{\cos^{2} (x^{2})}}$

Step 3.1.3.15

Multiply $- 1$ by $- 1$ .

$lim_{x \to 0^{-}} \frac{2 x}{\frac{\cos^{2} (x^{2}) (2 x) + 1 \sin (x^{2}) (\sin (x^{2}) \frac{d}{d x} [x^{2}])}{\cos^{2} (x^{2})}}$

Step 3.1.3.16

Multiply $\sin (x^{2})$ by $1$ .

$lim_{x \to 0^{-}} \frac{2 x}{\frac{\cos^{2} (x^{2}) (2 x) + \sin (x^{2}) (\sin (x^{2}) \frac{d}{d x} [x^{2}])}{\cos^{2} (x^{2})}}$

Step 3.1.3.17

Raise $\sin (x^{2})$ to the power of $1$ .

$lim_{x \to 0^{-}} \frac{2 x}{\frac{\cos^{2} (x^{2}) (2 x) + \sin^{1} (x^{2}) \sin (x^{2}) \frac{d}{d x} [x^{2}]}{\cos^{2} (x^{2})}}$

Step 3.1.3.18

Raise $\sin (x^{2})$ to the power of $1$ .

$lim_{x \to 0^{-}} \frac{2 x}{\frac{\cos^{2} (x^{2}) (2 x) + \sin^{1} (x^{2}) \sin^{1} (x^{2}) \frac{d}{d x} [x^{2}]}{\cos^{2} (x^{2})}}$

Step 3.1.3.19

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$lim_{x \to 0^{-}} \frac{2 x}{\frac{\cos^{2} (x^{2}) (2 x) + {\sin (x^{2})}^{1 + 1} \frac{d}{d x} [x^{2}]}{\cos^{2} (x^{2})}}$

Step 3.1.3.20

Add $1$ and $1$ .

$lim_{x \to 0^{-}} \frac{2 x}{\frac{\cos^{2} (x^{2}) (2 x) + \sin^{2} (x^{2}) \frac{d}{d x} [x^{2}]}{\cos^{2} (x^{2})}}$

Step 3.1.3.21

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$lim_{x \to 0^{-}} \frac{2 x}{\frac{\cos^{2} (x^{2}) (2 x) + \sin^{2} (x^{2}) (2 x)}{\cos^{2} (x^{2})}}$

Step 3.1.3.22

Simplify the numerator.

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Step 3.1.3.22.1

Factor $2 x$ out of $\cos^{2} (x^{2}) (2 x) + \sin^{2} (x^{2}) (2 x)$ .

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Step 3.1.3.22.1.1

Factor $2 x$ out of $\cos^{2} (x^{2}) (2 x)$ .

$lim_{x \to 0^{-}} \frac{2 x}{\frac{2 x (\cos^{2} (x^{2})) + \sin^{2} (x^{2}) (2 x)}{\cos^{2} (x^{2})}}$

Step 3.1.3.22.1.2

Factor $2 x$ out of $\sin^{2} (x^{2}) (2 x)$ .

$lim_{x \to 0^{-}} \frac{2 x}{\frac{2 x (\cos^{2} (x^{2})) + 2 x (\sin^{2} (x^{2}))}{\cos^{2} (x^{2})}}$

Step 3.1.3.22.1.3

Factor $2 x$ out of $2 x (\cos^{2} (x^{2})) + 2 x (\sin^{2} (x^{2}))$ .

$lim_{x \to 0^{-}} \frac{2 x}{\frac{2 x (\cos^{2} (x^{2}) + \sin^{2} (x^{2}))}{\cos^{2} (x^{2})}}$

Step 3.1.3.22.2

Rearrange terms.

$lim_{x \to 0^{-}} \frac{2 x}{\frac{2 x (\sin^{2} (x^{2}) + \cos^{2} (x^{2}))}{\cos^{2} (x^{2})}}$

Step 3.1.3.22.3

Apply pythagorean identity.

$lim_{x \to 0^{-}} \frac{2 x}{\frac{2 x \cdot 1}{\cos^{2} (x^{2})}}$

Step 3.1.3.22.4

Multiply $2$ by $1$ .

$lim_{x \to 0^{-}} \frac{2 x}{\frac{2 x}{\cos^{2} (x^{2})}}$

Step 3.1.4

Multiply the numerator by the reciprocal of the denominator.

$lim_{x \to 0^{-}} 2 x \frac{\cos^{2} (x^{2})}{2 x}$

Step 3.1.5

Combine factors.

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Step 3.1.5.1

Combine $2$ and $\frac{\cos^{2} (x^{2})}{2 x}$ .

$lim_{x \to 0^{-}} x \frac{2 \cos^{2} (x^{2})}{2 x}$

Step 3.1.5.2

Combine $x$ and $\frac{2 \cos^{2} (x^{2})}{2 x}$ .

$lim_{x \to 0^{-}} \frac{x (2 \cos^{2} (x^{2}))}{2 x}$

Step 3.1.6

Reduce.

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Step 3.1.6.1

Cancel the common factor of $x$ .

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Step 3.1.6.1.1

Cancel the common factor.

$lim_{x \to 0^{-}} \frac{x (2 \cos^{2} (x^{2}))}{2 x}$

Step 3.1.6.1.2

Rewrite the expression.

$lim_{x \to 0^{-}} \frac{2 \cos^{2} (x^{2})}{2}$

Step 3.1.6.2

Cancel the common factor of $2$ .

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Step 3.1.6.2.1

Cancel the common factor.

$lim_{x \to 0^{-}} \frac{2 \cos^{2} (x^{2})}{2}$

Step 3.1.6.2.2

Divide $\cos^{2} (x^{2})$ by $1$ .

$lim_{x \to 0^{-}} \cos^{2} (x^{2})$

Step 3.2

Evaluate the limit.

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Step 3.2.1

Move the exponent $2$ from $\cos^{2} (x^{2})$ outside the limit using the Limits Power Rule.

${(lim_{x \to 0^{-}} \cos (x^{2}))}^{2}$

Step 3.2.2

Move the limit inside the trig function because cosine is continuous.

$\cos^{2} (lim_{x \to 0^{-}} x^{2})$

Step 3.2.3

Move the exponent $2$ from $x^{2}$ outside the limit using the Limits Power Rule.

$\cos^{2} ({(lim_{x \to 0^{-}} x)}^{2})$

Step 3.3

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$\cos^{2} (0^{2})$

Step 3.4

Simplify the answer.

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Step 3.4.1

Raising $0$ to any positive power yields $0$ .

$\cos^{2} (0)$

Step 3.4.2

The exact value of $\cos (0)$ is $1$ .

$1^{2}$

Step 3.4.3

One to any power is one.

$1$

Step 4

Set up the limit as a right-sided limit.

$lim_{x \to 0^{+}} \frac{x^{2}}{\frac{1}{\cot (x^{2})}}$

Step 5

Evaluate the right-sided limit.

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Step 5.1

Apply L'Hospital's rule.

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Step 5.1.1

Evaluate the limit of the numerator and the limit of the denominator.

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Step 5.1.1.1

Take the limit of the numerator and the limit of the denominator.

$\frac{lim_{x \to 0^{+}} x^{2}}{lim_{x \to 0^{+}} \frac{1}{\cot (x^{2})}}$

Step 5.1.1.2

Evaluate the limit of the numerator.

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Step 5.1.1.2.1

Move the exponent $2$ from $x^{2}$ outside the limit using the Limits Power Rule.

$\frac{{(lim_{x \to 0^{+}} x)}^{2}}{lim_{x \to 0^{+}} \frac{1}{\cot (x^{2})}}$

Step 5.1.1.2.2

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$\frac{0^{2}}{lim_{x \to 0^{+}} \frac{1}{\cot (x^{2})}}$

Step 5.1.1.2.3

Raising $0$ to any positive power yields $0$ .

$\frac{0}{lim_{x \to 0^{+}} \frac{1}{\cot (x^{2})}}$

Step 5.1.1.3

Evaluate the limit of the denominator.

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Step 5.1.1.3.1

Apply trigonometric identities.

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Step 5.1.1.3.1.1

Rewrite $\cot (x^{2})$ in terms of sines and cosines.

$\frac{0}{lim_{x \to 0^{+}} \frac{1}{\frac{\cos (x^{2})}{\sin (x^{2})}}}$

Step 5.1.1.3.1.2

Multiply by the reciprocal of the fraction to divide by $\frac{\cos (x^{2})}{\sin (x^{2})}$ .

$\frac{0}{lim_{x \to 0^{+}} \frac{\sin (x^{2})}{\cos (x^{2})}}$

Step 5.1.1.3.1.3

Convert from $\frac{\sin (x^{2})}{\cos (x^{2})}$ to $\tan (x^{2})$ .

$\frac{0}{lim_{x \to 0^{+}} \tan (x^{2})}$

Step 5.1.1.3.2

Evaluate the limit.

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Step 5.1.1.3.2.1

Move the limit inside the trig function because tangent is continuous.

$\frac{0}{\tan (lim_{x \to 0^{+}} x^{2})}$

Step 5.1.1.3.2.2

Move the exponent $2$ from $x^{2}$ outside the limit using the Limits Power Rule.

$\frac{0}{\tan ({(lim_{x \to 0^{+}} x)}^{2})}$

Step 5.1.1.3.3

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$\frac{0}{\tan (0^{2})}$

Step 5.1.1.3.4

Simplify the answer.

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Step 5.1.1.3.4.1

Raising $0$ to any positive power yields $0$ .

$\frac{0}{\tan (0)}$

Step 5.1.1.3.4.2

The exact value of $\tan (0)$ is $0$ .

$\frac{0}{0}$

Step 5.1.1.3.4.3

The expression contains a division by $0$ . The expression is undefined.

Undefined

$\frac{0}{0}$

Step 5.1.1.3.5

The expression contains a division by $0$ . The expression is undefined.

Undefined

$\frac{0}{0}$

Step 5.1.1.4

The expression contains a division by $0$ . The expression is undefined.

Undefined

$\frac{0}{0}$

Step 5.1.2

Since $\frac{0}{0}$ is of indeterminate form, apply L'Hospital's Rule. L'Hospital's Rule states that the limit of a quotient of functions is equal to the limit of the quotient of their derivatives.

$lim_{x \to 0^{+}} \frac{x^{2}}{\frac{1}{\cot (x^{2})}} = lim_{x \to 0^{+}} \frac{\frac{d}{d x} [x^{2}]}{\frac{d}{d x} [\frac{1}{\cot (x^{2})}]}$

Step 5.1.3

Find the derivative of the numerator and denominator.

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Step 5.1.3.1

Differentiate the numerator and denominator.

$lim_{x \to 0^{+}} \frac{\frac{d}{d x} [x^{2}]}{\frac{d}{d x} [\frac{1}{\cot (x^{2})}]}$

Step 5.1.3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$lim_{x \to 0^{+}} \frac{2 x}{\frac{d}{d x} [\frac{1}{\cot (x^{2})}]}$

Step 5.1.3.3

Rewrite $\cot (x^{2})$ in terms of sines and cosines.

$lim_{x \to 0^{+}} \frac{2 x}{\frac{d}{d x} [\frac{1}{\frac{\cos (x^{2})}{\sin (x^{2})}}]}$

Step 5.1.3.4

Multiply by the reciprocal of the fraction to divide by $\frac{\cos (x^{2})}{\sin (x^{2})}$ .

$lim_{x \to 0^{+}} \frac{2 x}{\frac{d}{d x} [1 \frac{\sin (x^{2})}{\cos (x^{2})}]}$

Step 5.1.3.5

Write $1$ as a fraction with denominator $1$ .

$lim_{x \to 0^{+}} \frac{2 x}{\frac{d}{d x} [\frac{1}{1} \cdot \frac{\sin (x^{2})}{\cos (x^{2})}]}$

Step 5.1.3.6

Simplify.

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Step 5.1.3.6.1

Rewrite the expression.

$lim_{x \to 0^{+}} \frac{2 x}{\frac{d}{d x} [1 \frac{\sin (x^{2})}{\cos (x^{2})}]}$

Step 5.1.3.6.2

Multiply $\frac{\sin (x^{2})}{\cos (x^{2})}$ by $1$ .

$lim_{x \to 0^{+}} \frac{2 x}{\frac{d}{d x} [\frac{\sin (x^{2})}{\cos (x^{2})}]}$

Step 5.1.3.7

$lim_{x \to 0^{+}} \frac{2 x}{\frac{\cos (x^{2}) \frac{d}{d x} [\sin (x^{2})] - \sin (x^{2}) \frac{d}{d x} [\cos (x^{2})]}{\cos^{2} (x^{2})}}$

Step 5.1.3.8

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \sin (x)$ and $g (x) = x^{2}$ .

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Step 5.1.3.8.1

To apply the Chain Rule, set $u_{1}$ as $x^{2}$ .

$lim_{x \to 0^{+}} \frac{2 x}{\frac{\cos (x^{2}) (\frac{d}{d u_{1}} [\sin (u_{1})] \frac{d}{d x} [x^{2}]) - \sin (x^{2}) \frac{d}{d x} [\cos (x^{2})]}{\cos^{2} (x^{2})}}$

Step 5.1.3.8.2

The derivative of $\sin (u_{1})$ with respect to $u_{1}$ is $\cos (u_{1})$ .

$lim_{x \to 0^{+}} \frac{2 x}{\frac{\cos (x^{2}) (\cos (u_{1}) \frac{d}{d x} [x^{2}]) - \sin (x^{2}) \frac{d}{d x} [\cos (x^{2})]}{\cos^{2} (x^{2})}}$

Step 5.1.3.8.3

Replace all occurrences of $u_{1}$ with $x^{2}$ .

$lim_{x \to 0^{+}} \frac{2 x}{\frac{\cos (x^{2}) (\cos (x^{2}) \frac{d}{d x} [x^{2}]) - \sin (x^{2}) \frac{d}{d x} [\cos (x^{2})]}{\cos^{2} (x^{2})}}$

Step 5.1.3.9

Raise $\cos (x^{2})$ to the power of $1$ .

$lim_{x \to 0^{+}} \frac{2 x}{\frac{\cos^{1} (x^{2}) \cos (x^{2}) \frac{d}{d x} [x^{2}] - \sin (x^{2}) \frac{d}{d x} [\cos (x^{2})]}{\cos^{2} (x^{2})}}$

Step 5.1.3.10

Raise $\cos (x^{2})$ to the power of $1$ .

$lim_{x \to 0^{+}} \frac{2 x}{\frac{\cos^{1} (x^{2}) \cos^{1} (x^{2}) \frac{d}{d x} [x^{2}] - \sin (x^{2}) \frac{d}{d x} [\cos (x^{2})]}{\cos^{2} (x^{2})}}$

Step 5.1.3.11

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$lim_{x \to 0^{+}} \frac{2 x}{\frac{{\cos (x^{2})}^{1 + 1} \frac{d}{d x} [x^{2}] - \sin (x^{2}) \frac{d}{d x} [\cos (x^{2})]}{\cos^{2} (x^{2})}}$

Step 5.1.3.12

Add $1$ and $1$ .

$lim_{x \to 0^{+}} \frac{2 x}{\frac{\cos^{2} (x^{2}) \frac{d}{d x} [x^{2}] - \sin (x^{2}) \frac{d}{d x} [\cos (x^{2})]}{\cos^{2} (x^{2})}}$

Step 5.1.3.13

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$lim_{x \to 0^{+}} \frac{2 x}{\frac{\cos^{2} (x^{2}) (2 x) - \sin (x^{2}) \frac{d}{d x} [\cos (x^{2})]}{\cos^{2} (x^{2})}}$

Step 5.1.3.14

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \cos (x)$ and $g (x) = x^{2}$ .

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Step 5.1.3.14.1

To apply the Chain Rule, set $u_{2}$ as $x^{2}$ .

$lim_{x \to 0^{+}} \frac{2 x}{\frac{\cos^{2} (x^{2}) (2 x) - \sin (x^{2}) (\frac{d}{d u_{2}} [\cos (u_{2})] \frac{d}{d x} [x^{2}])}{\cos^{2} (x^{2})}}$

Step 5.1.3.14.2

The derivative of $\cos (u_{2})$ with respect to $u_{2}$ is $- \sin (u_{2})$ .

$lim_{x \to 0^{+}} \frac{2 x}{\frac{\cos^{2} (x^{2}) (2 x) - \sin (x^{2}) (- \sin (u_{2}) \frac{d}{d x} [x^{2}])}{\cos^{2} (x^{2})}}$

Step 5.1.3.14.3

Replace all occurrences of $u_{2}$ with $x^{2}$ .

$lim_{x \to 0^{+}} \frac{2 x}{\frac{\cos^{2} (x^{2}) (2 x) - \sin (x^{2}) (- \sin (x^{2}) \frac{d}{d x} [x^{2}])}{\cos^{2} (x^{2})}}$

Step 5.1.3.15

Multiply $- 1$ by $- 1$ .

$lim_{x \to 0^{+}} \frac{2 x}{\frac{\cos^{2} (x^{2}) (2 x) + 1 \sin (x^{2}) (\sin (x^{2}) \frac{d}{d x} [x^{2}])}{\cos^{2} (x^{2})}}$

Step 5.1.3.16

Multiply $\sin (x^{2})$ by $1$ .

$lim_{x \to 0^{+}} \frac{2 x}{\frac{\cos^{2} (x^{2}) (2 x) + \sin (x^{2}) (\sin (x^{2}) \frac{d}{d x} [x^{2}])}{\cos^{2} (x^{2})}}$

Step 5.1.3.17

Raise $\sin (x^{2})$ to the power of $1$ .

$lim_{x \to 0^{+}} \frac{2 x}{\frac{\cos^{2} (x^{2}) (2 x) + \sin^{1} (x^{2}) \sin (x^{2}) \frac{d}{d x} [x^{2}]}{\cos^{2} (x^{2})}}$

Step 5.1.3.18

Raise $\sin (x^{2})$ to the power of $1$ .

$lim_{x \to 0^{+}} \frac{2 x}{\frac{\cos^{2} (x^{2}) (2 x) + \sin^{1} (x^{2}) \sin^{1} (x^{2}) \frac{d}{d x} [x^{2}]}{\cos^{2} (x^{2})}}$

Step 5.1.3.19

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$lim_{x \to 0^{+}} \frac{2 x}{\frac{\cos^{2} (x^{2}) (2 x) + {\sin (x^{2})}^{1 + 1} \frac{d}{d x} [x^{2}]}{\cos^{2} (x^{2})}}$

Step 5.1.3.20

Add $1$ and $1$ .

$lim_{x \to 0^{+}} \frac{2 x}{\frac{\cos^{2} (x^{2}) (2 x) + \sin^{2} (x^{2}) \frac{d}{d x} [x^{2}]}{\cos^{2} (x^{2})}}$

Step 5.1.3.21

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$lim_{x \to 0^{+}} \frac{2 x}{\frac{\cos^{2} (x^{2}) (2 x) + \sin^{2} (x^{2}) (2 x)}{\cos^{2} (x^{2})}}$

Step 5.1.3.22

Simplify the numerator.

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Step 5.1.3.22.1

Factor $2 x$ out of $\cos^{2} (x^{2}) (2 x) + \sin^{2} (x^{2}) (2 x)$ .

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Step 5.1.3.22.1.1

Factor $2 x$ out of $\cos^{2} (x^{2}) (2 x)$ .

$lim_{x \to 0^{+}} \frac{2 x}{\frac{2 x (\cos^{2} (x^{2})) + \sin^{2} (x^{2}) (2 x)}{\cos^{2} (x^{2})}}$

Step 5.1.3.22.1.2

Factor $2 x$ out of $\sin^{2} (x^{2}) (2 x)$ .

$lim_{x \to 0^{+}} \frac{2 x}{\frac{2 x (\cos^{2} (x^{2})) + 2 x (\sin^{2} (x^{2}))}{\cos^{2} (x^{2})}}$

Step 5.1.3.22.1.3

Factor $2 x$ out of $2 x (\cos^{2} (x^{2})) + 2 x (\sin^{2} (x^{2}))$ .

$lim_{x \to 0^{+}} \frac{2 x}{\frac{2 x (\cos^{2} (x^{2}) + \sin^{2} (x^{2}))}{\cos^{2} (x^{2})}}$

Step 5.1.3.22.2

Rearrange terms.

$lim_{x \to 0^{+}} \frac{2 x}{\frac{2 x (\sin^{2} (x^{2}) + \cos^{2} (x^{2}))}{\cos^{2} (x^{2})}}$

Step 5.1.3.22.3

Apply pythagorean identity.

$lim_{x \to 0^{+}} \frac{2 x}{\frac{2 x \cdot 1}{\cos^{2} (x^{2})}}$

Step 5.1.3.22.4

Multiply $2$ by $1$ .

$lim_{x \to 0^{+}} \frac{2 x}{\frac{2 x}{\cos^{2} (x^{2})}}$

Step 5.1.4

Multiply the numerator by the reciprocal of the denominator.

$lim_{x \to 0^{+}} 2 x \frac{\cos^{2} (x^{2})}{2 x}$

Step 5.1.5

Combine factors.

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Step 5.1.5.1

Combine $2$ and $\frac{\cos^{2} (x^{2})}{2 x}$ .

$lim_{x \to 0^{+}} x \frac{2 \cos^{2} (x^{2})}{2 x}$

Step 5.1.5.2

Combine $x$ and $\frac{2 \cos^{2} (x^{2})}{2 x}$ .

$lim_{x \to 0^{+}} \frac{x (2 \cos^{2} (x^{2}))}{2 x}$

Step 5.1.6

Reduce.

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Step 5.1.6.1

Cancel the common factor of $x$ .

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Step 5.1.6.1.1

Cancel the common factor.

$lim_{x \to 0^{+}} \frac{x (2 \cos^{2} (x^{2}))}{2 x}$

Step 5.1.6.1.2

Rewrite the expression.

$lim_{x \to 0^{+}} \frac{2 \cos^{2} (x^{2})}{2}$

Step 5.1.6.2

Cancel the common factor of $2$ .

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Step 5.1.6.2.1

Cancel the common factor.

$lim_{x \to 0^{+}} \frac{2 \cos^{2} (x^{2})}{2}$

Step 5.1.6.2.2

Divide $\cos^{2} (x^{2})$ by $1$ .

$lim_{x \to 0^{+}} \cos^{2} (x^{2})$

Step 5.2

Evaluate the limit.

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Step 5.2.1

Move the exponent $2$ from $\cos^{2} (x^{2})$ outside the limit using the Limits Power Rule.

${(lim_{x \to 0^{+}} \cos (x^{2}))}^{2}$

Step 5.2.2

Move the limit inside the trig function because cosine is continuous.

$\cos^{2} (lim_{x \to 0^{+}} x^{2})$

Step 5.2.3

Move the exponent $2$ from $x^{2}$ outside the limit using the Limits Power Rule.

$\cos^{2} ({(lim_{x \to 0^{+}} x)}^{2})$

Step 5.3

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$\cos^{2} (0^{2})$

Step 5.4

Simplify the answer.

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Step 5.4.1

Raising $0$ to any positive power yields $0$ .

$\cos^{2} (0)$

Step 5.4.2

The exact value of $\cos (0)$ is $1$ .

$1^{2}$

Step 5.4.3

One to any power is one.

$1$

Step 6

Since the left-sided limit is equal to the right-sided limit, the limit is equal to $1$ .

$1$