Calculus Examples

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

Step 1

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

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Step 1.1

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

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

Step 1.2

Evaluate the limit of the numerator.

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Step 1.2.1

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

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

Step 1.2.2

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

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

Step 1.2.3

Move the limit inside the logarithm.

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

Step 1.2.4

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

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

Step 1.2.5

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

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

Step 1.2.6

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

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

Step 1.2.7

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

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Step 1.2.7.1

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

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

Step 1.2.7.2

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

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

Step 1.2.8

Simplify the answer.

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Step 1.2.8.1

Simplify each term.

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Step 1.2.8.1.1

Add $0$ and $1$ .

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

Step 1.2.8.1.2

The natural logarithm of $1$ is $0$ .

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

Step 1.2.8.1.3

Multiply $2$ by $0$ .

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

Step 1.2.8.1.4

Multiply $2$ by $0$ .

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

Step 1.2.8.2

Add $0$ and $0$ .

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

Step 1.3

Evaluate the limit of the denominator.

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Step 1.3.1

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

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

Step 1.3.2

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

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

Step 1.3.3

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

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

Step 1.3.4

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

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

Step 1.3.5

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

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

Step 1.3.6

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

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Step 1.3.6.1

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

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

Step 1.3.6.2

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

$\frac{0}{4 \sin (0) - 2 \cdot 0^{3}}$

Step 1.3.7

Simplify the answer.

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Step 1.3.7.1

Simplify each term.

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Step 1.3.7.1.1

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

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

Step 1.3.7.1.2

Multiply $4$ by $0$ .

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

Step 1.3.7.1.3

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

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

Step 1.3.7.1.4

Multiply $- 2$ by $0$ .

$\frac{0}{0 + 0}$

Step 1.3.7.2

Add $0$ and $0$ .

$\frac{0}{0}$

Step 1.3.7.3

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

Undefined

$\frac{0}{0}$

Step 1.3.8

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

Undefined

$\frac{0}{0}$

Step 1.4

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

Undefined

$\frac{0}{0}$

Step 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.

$lim_{x \to 0} \frac{2 \ln (x + 1) + 2 x}{4 \sin (x) - 2 x^{3}} = lim_{x \to 0} \frac{\frac{d}{d x} [2 \ln (x + 1) + 2 x]}{\frac{d}{d x} [4 \sin (x) - 2 x^{3}]}$

Step 3

Find the derivative of the numerator and denominator.

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Step 3.1

Differentiate the numerator and denominator.

$lim_{x \to 0} \frac{\frac{d}{d x} [2 \ln (x + 1) + 2 x]}{\frac{d}{d x} [4 \sin (x) - 2 x^{3}]}$

Step 3.2

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

$lim_{x \to 0} \frac{\frac{d}{d x} [2 \ln (x + 1)] + \frac{d}{d x} [2 x]}{\frac{d}{d x} [4 \sin (x) - 2 x^{3}]}$

Step 3.3

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

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Step 3.3.1

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

$lim_{x \to 0} \frac{2 \frac{d}{d x} [\ln (x + 1)] + \frac{d}{d x} [2 x]}{\frac{d}{d x} [4 \sin (x) - 2 x^{3}]}$

Step 3.3.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \ln (x)$ and $g (x) = x + 1$ .

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Step 3.3.2.1

To apply the Chain Rule, set $u$ as $x + 1$ .

$lim_{x \to 0} \frac{2 (\frac{d}{d u} [\ln (u)] \frac{d}{d x} [x + 1]) + \frac{d}{d x} [2 x]}{\frac{d}{d x} [4 \sin (x) - 2 x^{3}]}$

Step 3.3.2.2

The derivative of $\ln (u)$ with respect to $u$ is $\frac{1}{u}$ .

$lim_{x \to 0} \frac{2 (\frac{1}{u} \frac{d}{d x} [x + 1]) + \frac{d}{d x} [2 x]}{\frac{d}{d x} [4 \sin (x) - 2 x^{3}]}$

Step 3.3.2.3

Replace all occurrences of $u$ with $x + 1$ .

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

Step 3.3.3

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

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

Step 3.3.4

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

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

Step 3.3.5

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

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

Step 3.3.6

Add $1$ and $0$ .

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

Step 3.3.7

Multiply $\frac{1}{x + 1}$ by $1$ .

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

Step 3.3.8

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

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

Step 3.4

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

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Step 3.4.1

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

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

Step 3.4.2

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

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

Step 3.4.3

Multiply $2$ by $1$ .

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

Step 3.5

Simplify.

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Step 3.5.1

Combine terms.

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Step 3.5.1.1

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

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

Step 3.5.1.2

Combine the numerators over the common denominator.

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

Step 3.5.2

Reorder terms.

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

Step 3.6

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

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

Step 3.7

Evaluate $\frac{d}{d x} [4 \sin (x)]$ .

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Step 3.7.1

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

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

Step 3.7.2

The derivative of $\sin (x)$ with respect to $x$ is $\cos (x)$ .

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

Step 3.8

Evaluate $\frac{d}{d x} [- 2 x^{3}]$ .

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Step 3.8.1

Since $- 2$ is constant with respect to $x$ , the derivative of $- 2 x^{3}$ with respect to $x$ is $- 2 \frac{d}{d x} [x^{3}]$ .

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

Step 3.8.2

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

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

Step 3.8.3

Multiply $3$ by $- 2$ .

$lim_{x \to 0} \frac{\frac{2 (x + 1) + 2}{x + 1}}{4 \cos (x) - 6 x^{2}}$

Step 3.9

Reorder terms.

$lim_{x \to 0} \frac{\frac{2 (x + 1) + 2}{x + 1}}{- 6 x^{2} + 4 \cos (x)}$

Step 4

Multiply the numerator by the reciprocal of the denominator.

$lim_{x \to 0} \frac{2 (x + 1) + 2}{x + 1} \cdot \frac{1}{- 6 x^{2} + 4 \cos (x)}$

Step 5

Simplify terms.

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Step 5.1

Multiply $\frac{2 (x + 1) + 2}{x + 1}$ by $\frac{1}{- 6 x^{2} + 4 \cos (x)}$ .

$lim_{x \to 0} \frac{2 (x + 1) + 2}{(x + 1) (- 6 x^{2} + 4 \cos (x))}$

Step 5.2

Cancel the common factor of $2 (x + 1) + 2$ and $- 6 x^{2} + 4 \cos (x)$ .

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Step 5.2.1

Factor $2$ out of $2$ .

$lim_{x \to 0} \frac{2 (x + 1) + 2 (1)}{(x + 1) (- 6 x^{2} + 4 \cos (x))}$

Step 5.2.2

Factor $2$ out of $2 (x + 1) + 2 (1)$ .

$lim_{x \to 0} \frac{2 (x + 1 + 1)}{(x + 1) (- 6 x^{2} + 4 \cos (x))}$

Step 5.2.3

Cancel the common factors.

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Step 5.2.3.1

Factor $2$ out of $(x + 1) (- 6 x^{2} + 4 \cos (x))$ .

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

Step 5.2.3.2

Cancel the common factor.

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

Step 5.2.3.3

Rewrite the expression.

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

Step 6

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

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

Step 7

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

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

Step 8

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

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

Step 9

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

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

Step 10

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

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

Step 11

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

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

Step 12

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

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

Step 13

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

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

Step 14

Move the term $3$ outside of the limit because it is constant with respect to $x$ .

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

Step 15

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

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

Step 16

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

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

Step 17

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

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

Step 18

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

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Step 18.1

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

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

Step 18.2

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

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

Step 18.3

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

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

Step 18.4

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

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

Step 19

Simplify the answer.

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Step 19.1

Simplify the numerator.

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Step 19.1.1

Add $0$ and $1$ .

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

Step 19.1.2

Add $1$ and $1$ .

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

Step 19.2

Simplify the denominator.

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Step 19.2.1

Add $0$ and $1$ .

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

Step 19.2.2

Multiply $- 3 \cdot 0^{2} + 2 \cos (0)$ by $1$ .

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

Step 19.2.3

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

$\frac{2}{- 3 \cdot 0 + 2 \cos (0)}$

Step 19.2.4

Multiply $- 3$ by $0$ .

$\frac{2}{0 + 2 \cos (0)}$

Step 19.2.5

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

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

Step 19.2.6

Multiply $2$ by $1$ .

$\frac{2}{0 + 2}$

Step 19.2.7

Add $0$ and $2$ .

$\frac{2}{2}$

Step 19.3

Divide $2$ by $2$ .

$1$