Calculus Examples

$lim_{x \to 0} \frac{\arctan (x) - x}{x^{3}}$

Step 1

Apply L'Hospital's rule.

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Step 1.1

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

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Step 1.1.1

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

$\frac{lim_{x \to 0} \arctan (x) - x}{lim_{x \to 0} x^{3}}$

Step 1.1.2

Evaluate the limit of the numerator.

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Step 1.1.2.1

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $0$ .

$\frac{lim_{x \to 0} \arctan (x) - lim_{x \to 0} x}{lim_{x \to 0} x^{3}}$

Step 1.1.2.2

Evaluate the limits by plugging in $0$ for all occurrences of $x$ .

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Step 1.1.2.2.1

Evaluate the limit of $\arctan (x)$ by plugging in $0$ for $x$ .

$\frac{\arctan (0) - lim_{x \to 0} x}{lim_{x \to 0} x^{3}}$

Step 1.1.2.2.2

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

$\frac{0 - lim_{x \to 0} x}{lim_{x \to 0} x^{3}}$

Step 1.1.2.2.3

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

$\frac{0 + 0}{lim_{x \to 0} x^{3}}$

Step 1.1.2.3

Add $0$ and $0$ .

$\frac{0}{lim_{x \to 0} x^{3}}$

Step 1.1.3

Evaluate the limit of the denominator.

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Step 1.1.3.1

Move the exponent $3$ from $x^{3}$ outside the limit using the Limits Power Rule.

$\frac{0}{{(lim_{x \to 0} x)}^{3}}$

Step 1.1.3.2

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

$\frac{0}{0^{3}}$

Step 1.1.3.3

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

$\frac{0}{0}$

Step 1.1.3.4

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

Undefined

$\frac{0}{0}$

Step 1.1.4

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

Undefined

$\frac{0}{0}$

Step 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.

$lim_{x \to 0} \frac{\arctan (x) - x}{x^{3}} = lim_{x \to 0} \frac{\frac{d}{d x} [\arctan (x) - x]}{\frac{d}{d x} [x^{3}]}$

Step 1.3

Find the derivative of the numerator and denominator.

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Step 1.3.1

Differentiate the numerator and denominator.

$lim_{x \to 0} \frac{\frac{d}{d x} [\arctan (x) - x]}{\frac{d}{d x} [x^{3}]}$

Step 1.3.2

By the Sum Rule, the derivative of $\arctan (x) - x$ with respect to $x$ is $\frac{d}{d x} [\arctan (x)] + \frac{d}{d x} [- x]$ .

$lim_{x \to 0} \frac{\frac{d}{d x} [\arctan (x)] + \frac{d}{d x} [- x]}{\frac{d}{d x} [x^{3}]}$

Step 1.3.3

The derivative of $\arctan (x)$ with respect to $x$ is $\frac{1}{1 + x^{2}}$ .

$lim_{x \to 0} \frac{\frac{1}{1 + x^{2}} + \frac{d}{d x} [- x]}{\frac{d}{d x} [x^{3}]}$

Step 1.3.4

Evaluate $\frac{d}{d x} [- x]$ .

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Step 1.3.4.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- x$ with respect to $x$ is $- \frac{d}{d x} [x]$ .

$lim_{x \to 0} \frac{\frac{1}{1 + x^{2}} - \frac{d}{d x} [x]}{\frac{d}{d x} [x^{3}]}$

Step 1.3.4.2

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

$lim_{x \to 0} \frac{\frac{1}{1 + x^{2}} - 1 \cdot 1}{\frac{d}{d x} [x^{3}]}$

Step 1.3.4.3

Multiply $- 1$ by $1$ .

$lim_{x \to 0} \frac{\frac{1}{1 + x^{2}} - 1}{\frac{d}{d x} [x^{3}]}$

Step 1.3.5

Simplify.

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Step 1.3.5.1

Combine terms.

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Step 1.3.5.1.1

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{1 + x^{2}}{1 + x^{2}}$ .

$lim_{x \to 0} \frac{\frac{1}{1 + x^{2}} - 1 \cdot \frac{1 + x^{2}}{1 + x^{2}}}{\frac{d}{d x} [x^{3}]}$

Step 1.3.5.1.2

Combine $- 1$ and $\frac{1 + x^{2}}{1 + x^{2}}$ .

$lim_{x \to 0} \frac{\frac{1}{1 + x^{2}} + \frac{- (1 + x^{2})}{1 + x^{2}}}{\frac{d}{d x} [x^{3}]}$

Step 1.3.5.1.3

Combine the numerators over the common denominator.

$lim_{x \to 0} \frac{\frac{1 - (1 + x^{2})}{1 + x^{2}}}{\frac{d}{d x} [x^{3}]}$

Step 1.3.5.2

Reorder terms.

$lim_{x \to 0} \frac{\frac{1 - (1 + x^{2})}{x^{2} + 1}}{\frac{d}{d x} [x^{3}]}$

Step 1.3.6

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

$lim_{x \to 0} \frac{\frac{1 - (1 + x^{2})}{x^{2} + 1}}{3 x^{2}}$

Step 1.4

Multiply the numerator by the reciprocal of the denominator.

$lim_{x \to 0} \frac{1 - (1 + x^{2})}{x^{2} + 1} \cdot \frac{1}{3 x^{2}}$

Step 1.5

Multiply $\frac{1 - (1 + x^{2})}{x^{2} + 1}$ by $\frac{1}{3 x^{2}}$ .

$lim_{x \to 0} \frac{1 - (1 + x^{2})}{(x^{2} + 1) (3 x^{2})}$

Step 2

Move the term $\frac{1}{3}$ outside of the limit because it is constant with respect to $x$ .

$\frac{1}{3} lim_{x \to 0} \frac{1 - (1 + x^{2})}{(x^{2} + 1) (x^{2})}$

Step 3

Apply L'Hospital's rule.

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Step 3.1

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

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Step 3.1.1

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

$\frac{1}{3} \cdot \frac{lim_{x \to 0} 1 - (1 + x^{2})}{lim_{x \to 0} (x^{2} + 1) (x^{2})}$

Step 3.1.2

Evaluate the limits by plugging in $0$ for all occurrences of $x$ .

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Step 3.1.2.1

Evaluate the limit of $1 - (1 + x^{2})$ by plugging in $0$ for $x$ .

$\frac{1}{3} \cdot \frac{1 - (1 + 0^{2})}{lim_{x \to 0} (x^{2} + 1) (x^{2})}$

Step 3.1.2.2

Simplify each term.

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Step 3.1.2.2.1

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

$\frac{1}{3} \cdot \frac{1 - (1 + 0)}{lim_{x \to 0} (x^{2} + 1) (x^{2})}$

Step 3.1.2.2.2

Add $1$ and $0$ .

$\frac{1}{3} \cdot \frac{1 - 1 \cdot 1}{lim_{x \to 0} (x^{2} + 1) (x^{2})}$

Step 3.1.2.2.3

Multiply $- 1$ by $1$ .

$\frac{1}{3} \cdot \frac{1 - 1}{lim_{x \to 0} (x^{2} + 1) (x^{2})}$

Step 3.1.2.3

Subtract $1$ from $1$ .

$\frac{1}{3} \cdot \frac{0}{lim_{x \to 0} (x^{2} + 1) (x^{2})}$

Step 3.1.3

Evaluate the limit of the denominator.

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Step 3.1.3.1

Split the limit using the Product of Limits Rule on the limit as $x$ approaches $0$ .

$\frac{1}{3} \cdot \frac{0}{lim_{x \to 0} x^{2} + 1 \cdot lim_{x \to 0} x^{2}}$

Step 3.1.3.2

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $0$ .

$\frac{1}{3} \cdot \frac{0}{(lim_{x \to 0} x^{2} + lim_{x \to 0} 1) \cdot lim_{x \to 0} x^{2}}$

Step 3.1.3.3

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

$\frac{1}{3} \cdot \frac{0}{({(lim_{x \to 0} x)}^{2} + lim_{x \to 0} 1) \cdot lim_{x \to 0} x^{2}}$

Step 3.1.3.4

Evaluate the limit of $1$ which is constant as $x$ approaches $0$ .

$\frac{1}{3} \cdot \frac{0}{({(lim_{x \to 0} x)}^{2} + 1) \cdot lim_{x \to 0} x^{2}}$

Step 3.1.3.5

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

$\frac{1}{3} \cdot \frac{0}{({(lim_{x \to 0} x)}^{2} + 1) \cdot {(lim_{x \to 0} x)}^{2}}$

Step 3.1.3.6

Evaluate the limits by plugging in $0$ for all occurrences of $x$ .

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Step 3.1.3.6.1

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

$\frac{1}{3} \cdot \frac{0}{(0^{2} + 1) \cdot {(lim_{x \to 0} x)}^{2}}$

Step 3.1.3.6.2

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

$\frac{1}{3} \cdot \frac{0}{(0^{2} + 1) \cdot 0^{2}}$

Step 3.1.3.7

Simplify the answer.

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Step 3.1.3.7.1

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

$\frac{1}{3} \cdot \frac{0}{(0 + 1) \cdot 0^{2}}$

Step 3.1.3.7.2

Add $0$ and $1$ .

$\frac{1}{3} \cdot \frac{0}{1 \cdot 0^{2}}$

Step 3.1.3.7.3

Multiply $0^{2}$ by $1$ .

$\frac{1}{3} \cdot \frac{0}{0^{2}}$

Step 3.1.3.7.4

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

$\frac{1}{3} \cdot \frac{0}{0}$

Step 3.1.3.7.5

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

Undefined

$\frac{1}{3} \cdot \frac{0}{0}$

Step 3.1.3.8

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

Undefined

$\frac{1}{3} \cdot \frac{0}{0}$

Step 3.1.4

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

Undefined

$\frac{1}{3} \cdot \frac{0}{0}$

Step 3.2

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.

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

Step 3.3

Find the derivative of the numerator and denominator.

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Step 3.3.1

Differentiate the numerator and denominator.

$\frac{1}{3} lim_{x \to 0} \frac{\frac{d}{d x} [1 - (1 + x^{2})]}{\frac{d}{d x} [(x^{2} + 1) (x^{2})]}$

Step 3.3.2

By the Sum Rule, the derivative of $1 - (1 + x^{2})$ with respect to $x$ is $\frac{d}{d x} [1] + \frac{d}{d x} [- (1 + x^{2})]$ .

$\frac{1}{3} lim_{x \to 0} \frac{\frac{d}{d x} [1] + \frac{d}{d x} [- (1 + x^{2})]}{\frac{d}{d x} [(x^{2} + 1) (x^{2})]}$

Step 3.3.3

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$\frac{1}{3} lim_{x \to 0} \frac{0 + \frac{d}{d x} [- (1 + x^{2})]}{\frac{d}{d x} [(x^{2} + 1) (x^{2})]}$

Step 3.3.4

Evaluate $\frac{d}{d x} [- (1 + x^{2})]$ .

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Step 3.3.4.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- (1 + x^{2})$ with respect to $x$ is $- \frac{d}{d x} [1 + x^{2}]$ .

$\frac{1}{3} lim_{x \to 0} \frac{0 - \frac{d}{d x} [1 + x^{2}]}{\frac{d}{d x} [(x^{2} + 1) (x^{2})]}$

Step 3.3.4.2

By the Sum Rule, the derivative of $1 + x^{2}$ with respect to $x$ is $\frac{d}{d x} [1] + \frac{d}{d x} [x^{2}]$ .

$\frac{1}{3} lim_{x \to 0} \frac{0 - (\frac{d}{d x} [1] + \frac{d}{d x} [x^{2}])}{\frac{d}{d x} [(x^{2} + 1) (x^{2})]}$

Step 3.3.4.3

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$\frac{1}{3} lim_{x \to 0} \frac{0 - (0 + \frac{d}{d x} [x^{2}])}{\frac{d}{d x} [(x^{2} + 1) (x^{2})]}$

Step 3.3.4.4

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

$\frac{1}{3} lim_{x \to 0} \frac{0 - (0 + 2 x)}{\frac{d}{d x} [(x^{2} + 1) (x^{2})]}$

Step 3.3.4.5

Add $0$ and $2 x$ .

$\frac{1}{3} lim_{x \to 0} \frac{0 - 1 \cdot 2 x}{\frac{d}{d x} [(x^{2} + 1) (x^{2})]}$

Step 3.3.4.6

Multiply $2$ by $- 1$ .

$\frac{1}{3} lim_{x \to 0} \frac{0 - 2 x}{\frac{d}{d x} [(x^{2} + 1) (x^{2})]}$

Step 3.3.5

Subtract $2 x$ from $0$ .

$\frac{1}{3} lim_{x \to 0} \frac{- 2 x}{\frac{d}{d x} [(x^{2} + 1) (x^{2})]}$

Step 3.3.6

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x^{2} + 1$ and $g (x) = x^{2}$ .

$\frac{1}{3} lim_{x \to 0} \frac{- 2 x}{(x^{2} + 1) \frac{d}{d x} [x^{2}] + x^{2} \frac{d}{d x} [x^{2} + 1]}$

Step 3.3.7

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

$\frac{1}{3} lim_{x \to 0} \frac{- 2 x}{(x^{2} + 1) \cdot 2 x + x^{2} \frac{d}{d x} [x^{2} + 1]}$

Step 3.3.8

Move $2$ to the left of $x^{2} + 1$ .

$\frac{1}{3} lim_{x \to 0} \frac{- 2 x}{2 \cdot (x^{2} + 1) x + x^{2} \frac{d}{d x} [x^{2} + 1]}$

Step 3.3.9

By the Sum Rule, the derivative of $x^{2} + 1$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [1]$ .

$\frac{1}{3} lim_{x \to 0} \frac{- 2 x}{2 (x^{2} + 1) x + x^{2} (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [1])}$

Step 3.3.10

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

$\frac{1}{3} lim_{x \to 0} \frac{- 2 x}{2 (x^{2} + 1) x + x^{2} (2 x + \frac{d}{d x} [1])}$

Step 3.3.11

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$\frac{1}{3} lim_{x \to 0} \frac{- 2 x}{2 (x^{2} + 1) x + x^{2} (2 x + 0)}$

Step 3.3.12

Add $2 x$ and $0$ .

$\frac{1}{3} lim_{x \to 0} \frac{- 2 x}{2 (x^{2} + 1) x + x^{2} \cdot 2 x}$

Step 3.3.13

Multiply $x^{2}$ by $x$ by adding the exponents.

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Step 3.3.13.1

Move $x$ .

$\frac{1}{3} lim_{x \to 0} \frac{- 2 x}{2 (x^{2} + 1) x + x \cdot x^{2} \cdot 2}$

Step 3.3.13.2

Multiply $x$ by $x^{2}$ .

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Step 3.3.13.2.1

Raise $x$ to the power of $1$ .

$\frac{1}{3} lim_{x \to 0} \frac{- 2 x}{2 (x^{2} + 1) x + x^{1} x^{2} \cdot 2}$

Step 3.3.13.2.2

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

$\frac{1}{3} lim_{x \to 0} \frac{- 2 x}{2 (x^{2} + 1) x + x^{1 + 2} \cdot 2}$

Step 3.3.13.3

Add $1$ and $2$ .

$\frac{1}{3} lim_{x \to 0} \frac{- 2 x}{2 (x^{2} + 1) x + x^{3} \cdot 2}$

Step 3.3.14

Move $2$ to the left of $x^{3}$ .

$\frac{1}{3} lim_{x \to 0} \frac{- 2 x}{2 (x^{2} + 1) x + 2 \cdot x^{3}}$

Step 3.3.15

Simplify.

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Step 3.3.15.1

Apply the distributive property.

$\frac{1}{3} lim_{x \to 0} \frac{- 2 x}{(2 x^{2} + 2 \cdot 1) x + 2 x^{3}}$

Step 3.3.15.2

Apply the distributive property.

$\frac{1}{3} lim_{x \to 0} \frac{- 2 x}{2 x^{2} x + 2 \cdot 1 x + 2 x^{3}}$

Step 3.3.15.3

Combine terms.

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Step 3.3.15.3.1

Raise $x$ to the power of $1$ .

$\frac{1}{3} lim_{x \to 0} \frac{- 2 x}{2 x^{1} x^{2} + 2 \cdot 1 x + 2 x^{3}}$

Step 3.3.15.3.2

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

$\frac{1}{3} lim_{x \to 0} \frac{- 2 x}{2 x^{1 + 2} + 2 \cdot 1 x + 2 x^{3}}$

Step 3.3.15.3.3

Add $1$ and $2$ .

$\frac{1}{3} lim_{x \to 0} \frac{- 2 x}{2 x^{3} + 2 \cdot 1 x + 2 x^{3}}$

Step 3.3.15.3.4

Multiply $2$ by $1$ .

$\frac{1}{3} lim_{x \to 0} \frac{- 2 x}{2 x^{3} + 2 x + 2 x^{3}}$

Step 3.3.15.3.5

Add $2 x^{3}$ and $2 x^{3}$ .

$\frac{1}{3} lim_{x \to 0} \frac{- 2 x}{4 x^{3} + 2 x}$

Step 4

Move the term $- 2$ outside of the limit because it is constant with respect to $x$ .

$\frac{1}{3} \cdot - 2 lim_{x \to 0} \frac{x}{4 x^{3} + 2 x}$

Step 5

Apply L'Hospital's rule.

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Step 5.1

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

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Step 5.1.1

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

$\frac{1}{3} \cdot - 2 \frac{lim_{x \to 0} x}{lim_{x \to 0} 4 x^{3} + 2 x}$

Step 5.1.2

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

$\frac{1}{3} \cdot - 2 \frac{0}{lim_{x \to 0} 4 x^{3} + 2 x}$

Step 5.1.3

Evaluate the limit of the denominator.

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Step 5.1.3.1

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $0$ .

$\frac{1}{3} \cdot - 2 \frac{0}{lim_{x \to 0} 4 x^{3} + lim_{x \to 0} 2 x}$

Step 5.1.3.2

Move the term $4$ outside of the limit because it is constant with respect to $x$ .

$\frac{1}{3} \cdot - 2 \frac{0}{4 lim_{x \to 0} x^{3} + lim_{x \to 0} 2 x}$

Step 5.1.3.3

Move the exponent $3$ from $x^{3}$ outside the limit using the Limits Power Rule.

$\frac{1}{3} \cdot - 2 \frac{0}{4 {(lim_{x \to 0} x)}^{3} + lim_{x \to 0} 2 x}$

Step 5.1.3.4

Move the term $2$ outside of the limit because it is constant with respect to $x$ .

$\frac{1}{3} \cdot - 2 \frac{0}{4 {(lim_{x \to 0} x)}^{3} + 2 lim_{x \to 0} x}$

Step 5.1.3.5

Evaluate the limits by plugging in $0$ for all occurrences of $x$ .

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Step 5.1.3.5.1

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

$\frac{1}{3} \cdot - 2 \frac{0}{4 \cdot 0^{3} + 2 lim_{x \to 0} x}$

Step 5.1.3.5.2

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

$\frac{1}{3} \cdot - 2 \frac{0}{4 \cdot 0^{3} + 2 \cdot 0}$

Step 5.1.3.6

Simplify the answer.

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Step 5.1.3.6.1

Simplify each term.

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Step 5.1.3.6.1.1

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

$\frac{1}{3} \cdot - 2 \frac{0}{4 \cdot 0 + 2 \cdot 0}$

Step 5.1.3.6.1.2

Multiply $4$ by $0$ .

$\frac{1}{3} \cdot - 2 \frac{0}{0 + 2 \cdot 0}$

Step 5.1.3.6.1.3

Multiply $2$ by $0$ .

$\frac{1}{3} \cdot - 2 \frac{0}{0 + 0}$

Step 5.1.3.6.2

Add $0$ and $0$ .

$\frac{1}{3} \cdot - 2 \frac{0}{0}$

Step 5.1.3.6.3

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

Undefined

$\frac{1}{3} \cdot - 2 \frac{0}{0}$

Step 5.1.3.7

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

Undefined

$\frac{1}{3} \cdot - 2 \frac{0}{0}$

Step 5.1.4

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

Undefined

$\frac{1}{3} \cdot - 2 \frac{0}{0}$

Step 5.2

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.

$lim_{x \to 0} \frac{x}{4 x^{3} + 2 x} = lim_{x \to 0} \frac{\frac{d}{d x} [x]}{\frac{d}{d x} [4 x^{3} + 2 x]}$

Step 5.3

Find the derivative of the numerator and denominator.

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Step 5.3.1

Differentiate the numerator and denominator.

$\frac{1}{3} \cdot - 2 lim_{x \to 0} \frac{\frac{d}{d x} [x]}{\frac{d}{d x} [4 x^{3} + 2 x]}$

Step 5.3.2

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

$\frac{1}{3} \cdot - 2 lim_{x \to 0} \frac{1}{\frac{d}{d x} [4 x^{3} + 2 x]}$

Step 5.3.3

By the Sum Rule, the derivative of $4 x^{3} + 2 x$ with respect to $x$ is $\frac{d}{d x} [4 x^{3}] + \frac{d}{d x} [2 x]$ .

$\frac{1}{3} \cdot - 2 lim_{x \to 0} \frac{1}{\frac{d}{d x} [4 x^{3}] + \frac{d}{d x} [2 x]}$

Step 5.3.4

Evaluate $\frac{d}{d x} [4 x^{3}]$ .

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Step 5.3.4.1

Since $4$ is constant with respect to $x$ , the derivative of $4 x^{3}$ with respect to $x$ is $4 \frac{d}{d x} [x^{3}]$ .

$\frac{1}{3} \cdot - 2 lim_{x \to 0} \frac{1}{4 \frac{d}{d x} [x^{3}] + \frac{d}{d x} [2 x]}$

Step 5.3.4.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

$\frac{1}{3} \cdot - 2 lim_{x \to 0} \frac{1}{4 \cdot 3 x^{2} + \frac{d}{d x} [2 x]}$

Step 5.3.4.3

Multiply $3$ by $4$ .

$\frac{1}{3} \cdot - 2 lim_{x \to 0} \frac{1}{12 x^{2} + \frac{d}{d x} [2 x]}$

Step 5.3.5

Evaluate $\frac{d}{d x} [2 x]$ .

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Step 5.3.5.1

Since $2$ is constant with respect to $x$ , the derivative of $2 x$ with respect to $x$ is $2 \frac{d}{d x} [x]$ .

$\frac{1}{3} \cdot - 2 lim_{x \to 0} \frac{1}{12 x^{2} + 2 \frac{d}{d x} [x]}$

Step 5.3.5.2

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

$\frac{1}{3} \cdot - 2 lim_{x \to 0} \frac{1}{12 x^{2} + 2 \cdot 1}$

Step 5.3.5.3

Multiply $2$ by $1$ .

$\frac{1}{3} \cdot - 2 lim_{x \to 0} \frac{1}{12 x^{2} + 2}$

Step 6

Evaluate the limit.

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Step 6.1

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $0$ .

$\frac{1}{3} \cdot - 2 \frac{lim_{x \to 0} 1}{lim_{x \to 0} 12 x^{2} + 2}$

Step 6.2

Evaluate the limit of $1$ which is constant as $x$ approaches $0$ .

$\frac{1}{3} \cdot - 2 \frac{1}{lim_{x \to 0} 12 x^{2} + 2}$

Step 6.3

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $0$ .

$\frac{1}{3} \cdot - 2 \frac{1}{lim_{x \to 0} 12 x^{2} + lim_{x \to 0} 2}$

Step 6.4

Move the term $12$ outside of the limit because it is constant with respect to $x$ .

$\frac{1}{3} \cdot - 2 \frac{1}{12 lim_{x \to 0} x^{2} + lim_{x \to 0} 2}$

Step 6.5

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

$\frac{1}{3} \cdot - 2 \frac{1}{12 {(lim_{x \to 0} x)}^{2} + lim_{x \to 0} 2}$

Step 6.6

Evaluate the limit of $2$ which is constant as $x$ approaches $0$ .

$\frac{1}{3} \cdot - 2 \frac{1}{12 {(lim_{x \to 0} x)}^{2} + 2}$

Step 7

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

$\frac{1}{3} \cdot - 2 \frac{1}{12 \cdot 0^{2} + 2}$

Step 8

Simplify the answer.

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Step 8.1

Combine $\frac{1}{3}$ and $- 2$ .

$\frac{- 2}{3} \cdot \frac{1}{12 \cdot 0^{2} + 2}$

Step 8.2

Move the negative in front of the fraction.

$- \frac{2}{3} \cdot \frac{1}{12 \cdot 0^{2} + 2}$

Step 8.3

Simplify the denominator.

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Step 8.3.1

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

$- \frac{2}{3} \cdot \frac{1}{12 \cdot 0 + 2}$

Step 8.3.2

Multiply $12$ by $0$ .

$- \frac{2}{3} \cdot \frac{1}{0 + 2}$

Step 8.3.3

Add $0$ and $2$ .

$- \frac{2}{3} \cdot \frac{1}{2}$

Step 8.4

Cancel the common factor of $2$ .

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Step 8.4.1

Move the leading negative in $- \frac{2}{3}$ into the numerator.

$\frac{- 2}{3} \cdot \frac{1}{2}$

Step 8.4.2

Factor $2$ out of $- 2$ .

$\frac{2 (- 1)}{3} \cdot \frac{1}{2}$

Step 8.4.3

Cancel the common factor.

$\frac{2 \cdot - 1}{3} \cdot \frac{1}{2}$

Step 8.4.4

Rewrite the expression.

$\frac{- 1}{3}$

Step 8.5

Move the negative in front of the fraction.

$- \frac{1}{3}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$- \frac{1}{3}$

Decimal Form:

$- 0.\overline{3}$