Calculus Examples

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

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.

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

Step 1.1.2

Evaluate the limit of the numerator.

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Step 1.1.2.1

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

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

Step 1.1.2.2

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

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

Step 1.1.2.3

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

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

Step 1.1.2.4

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

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

Step 1.1.2.5

Move the term $- 1$ outside of the limit because it is constant with respect to $x$ .

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

Step 1.1.2.6

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

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Step 1.1.2.6.1

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

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

Step 1.1.2.6.2

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

$\frac{\sin (4 \cdot 0) \cdot \sin (0)}{lim_{x \to 0} x \sin (3 x)}$

Step 1.1.2.7

Simplify the answer.

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Step 1.1.2.7.1

Multiply $4$ by $0$ .

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

Step 1.1.2.7.2

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

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

Step 1.1.2.7.3

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

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

Step 1.1.2.7.4

Multiply $0$ by $0$ .

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

Step 1.1.3

Evaluate the limit of the denominator.

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Step 1.1.3.1

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

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

Step 1.1.3.2

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

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

Step 1.1.3.3

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

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

Step 1.1.3.4

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

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Step 1.1.3.4.1

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

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

Step 1.1.3.4.2

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

$\frac{0}{0 \cdot \sin (3 \cdot 0)}$

Step 1.1.3.5

Simplify the answer.

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Step 1.1.3.5.1

Multiply $3$ by $0$ .

$\frac{0}{0 \cdot \sin (0)}$

Step 1.1.3.5.2

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

$\frac{0}{0 \cdot 0}$

Step 1.1.3.5.3

Multiply $0$ by $0$ .

$\frac{0}{0}$

Step 1.1.3.5.4

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

Undefined

$\frac{0}{0}$

Step 1.1.3.6

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.

$lim_{x \to 0} \frac{\sin (4 x) \sin (- x)}{x \sin (3 x)} = lim_{x \to 0} \frac{\frac{d}{d x} [\sin (4 x) \sin (- x)]}{\frac{d}{d x} [x \sin (3 x)]}$

Step 1.3

Find the derivative of the numerator and denominator.

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Step 1.3.1

Differentiate the numerator and denominator.

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

Step 1.3.2

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 $f (x) = \sin (4 x)$ and $g (x) = \sin (- x)$ .

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

Step 1.3.3

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 $g (x) = - x$ .

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Step 1.3.3.1

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

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

Step 1.3.3.2

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

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

Step 1.3.3.3

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

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

Step 1.3.4

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

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

Step 1.3.5

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

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

Step 1.3.6

Multiply $- 1$ by $1$ .

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

Step 1.3.7

Move $- 1$ to the left of $\sin (4 x) \cos (- x)$ .

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

Step 1.3.8

Rewrite $- 1 \sin (4 x)$ as $- \sin (4 x)$ .

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

Step 1.3.9

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 $g (x) = 4 x$ .

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Step 1.3.9.1

To apply the Chain Rule, set $u_{2}$ as $4 x$ .

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

Step 1.3.9.2

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

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

Step 1.3.9.3

Replace all occurrences of $u_{2}$ with $4 x$ .

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

Step 1.3.10

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

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

Step 1.3.11

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

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

Step 1.3.12

Multiply $4$ by $1$ .

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

Step 1.3.13

Move $4$ to the left of $\sin (- x) \cos (4 x)$ .

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

Step 1.3.14

Simplify.

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Step 1.3.14.1

Reorder terms.

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

Step 1.3.14.2

Simplify each term.

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Step 1.3.14.2.1

Since $\cos (- x)$ is an even function, rewrite $\cos (- x)$ as $\cos (x)$ .

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

Step 1.3.14.2.2

Since $\sin (- x)$ is an odd function, rewrite $\sin (- x)$ as $- \sin (x)$ .

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

Step 1.3.14.2.3

Multiply $- 1$ by $4$ .

$lim_{x \to 0} \frac{- \cos (x) \sin (4 x) - 4 \cos (4 x) \sin (x)}{\frac{d}{d x} [x \sin (3 x)]}$

Step 1.3.15

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 $f (x) = x$ and $g (x) = \sin (3 x)$ .

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

Step 1.3.16

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 $g (x) = 3 x$ .

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Step 1.3.16.1

To apply the Chain Rule, set $u$ as $3 x$ .

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

Step 1.3.16.2

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

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

Step 1.3.16.3

Replace all occurrences of $u$ with $3 x$ .

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

Step 1.3.17

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

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

Step 1.3.18

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

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

Step 1.3.19

Multiply $3$ by $1$ .

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

Step 1.3.20

Move $3$ to the left of $\cos (3 x)$ .

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

Step 1.3.21

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

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

Step 1.3.22

Multiply $\sin (3 x)$ by $1$ .

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

Step 1.3.23

Reorder terms.

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

Step 2

Apply L'Hospital's rule.

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Step 2.1

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

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Step 2.1.1

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

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

Step 2.1.2

Evaluate the limit of the numerator.

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Step 2.1.2.1

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

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

Step 2.1.2.2

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

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

Step 2.1.2.3

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

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

Step 2.1.2.4

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

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

Step 2.1.2.5

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

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

Step 2.1.2.6

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

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

Step 2.1.2.7

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

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

Step 2.1.2.8

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

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

Step 2.1.2.9

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

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

Step 2.1.2.10

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

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

Step 2.1.2.11

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

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Step 2.1.2.11.1

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

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

Step 2.1.2.11.2

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

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

Step 2.1.2.11.3

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

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

Step 2.1.2.11.4

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

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

Step 2.1.2.12

Simplify the answer.

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Step 2.1.2.12.1

Simplify each term.

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Step 2.1.2.12.1.1

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

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

Step 2.1.2.12.1.2

Multiply $- 1$ by $1$ .

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

Step 2.1.2.12.1.3

Multiply $4$ by $0$ .

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

Step 2.1.2.12.1.4

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

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

Step 2.1.2.12.1.5

Multiply $- 1$ by $0$ .

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

Step 2.1.2.12.1.6

Multiply $4$ by $0$ .

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

Step 2.1.2.12.1.7

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

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

Step 2.1.2.12.1.8

Multiply $- 4$ by $1$ .

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

Step 2.1.2.12.1.9

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

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

Step 2.1.2.12.1.10

Multiply $- 4$ by $0$ .

$\frac{0 + 0}{lim_{x \to 0} 3 x \cos (3 x) + \sin (3 x)}$

Step 2.1.2.12.2

Add $0$ and $0$ .

$\frac{0}{lim_{x \to 0} 3 x \cos (3 x) + \sin (3 x)}$

Step 2.1.3

Evaluate the limit of the denominator.

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Step 2.1.3.1

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

$\frac{0}{lim_{x \to 0} 3 x \cos (3 x) + lim_{x \to 0} \sin (3 x)}$

Step 2.1.3.2

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

$\frac{0}{3 lim_{x \to 0} x \cos (3 x) + lim_{x \to 0} \sin (3 x)}$

Step 2.1.3.3

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

$\frac{0}{3 lim_{x \to 0} x \cdot lim_{x \to 0} \cos (3 x) + lim_{x \to 0} \sin (3 x)}$

Step 2.1.3.4

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

$\frac{0}{3 lim_{x \to 0} x \cdot \cos (lim_{x \to 0} 3 x) + lim_{x \to 0} \sin (3 x)}$

Step 2.1.3.5

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

$\frac{0}{3 lim_{x \to 0} x \cdot \cos (3 lim_{x \to 0} x) + lim_{x \to 0} \sin (3 x)}$

Step 2.1.3.6

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

$\frac{0}{3 lim_{x \to 0} x \cdot \cos (3 lim_{x \to 0} x) + \sin (lim_{x \to 0} 3 x)}$

Step 2.1.3.7

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

$\frac{0}{3 lim_{x \to 0} x \cdot \cos (3 lim_{x \to 0} x) + \sin (3 lim_{x \to 0} x)}$

Step 2.1.3.8

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

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Step 2.1.3.8.1

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

$\frac{0}{3 \cdot 0 \cdot \cos (3 lim_{x \to 0} x) + \sin (3 lim_{x \to 0} x)}$

Step 2.1.3.8.2

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

$\frac{0}{3 \cdot 0 \cdot \cos (3 \cdot 0) + \sin (3 lim_{x \to 0} x)}$

Step 2.1.3.8.3

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

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

Step 2.1.3.9

Simplify the answer.

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Step 2.1.3.9.1

Simplify each term.

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Step 2.1.3.9.1.1

Multiply $3$ by $0$ .

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

Step 2.1.3.9.1.2

Multiply $3$ by $0$ .

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

Step 2.1.3.9.1.3

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

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

Step 2.1.3.9.1.4

Multiply $0$ by $1$ .

$\frac{0}{0 + \sin (3 \cdot 0)}$

Step 2.1.3.9.1.5

Multiply $3$ by $0$ .

$\frac{0}{0 + \sin (0)}$

Step 2.1.3.9.1.6

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

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

Step 2.1.3.9.2

Add $0$ and $0$ .

$\frac{0}{0}$

Step 2.1.3.9.3

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

Undefined

$\frac{0}{0}$

Step 2.1.3.10

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

Undefined

$\frac{0}{0}$

Step 2.1.4

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

Undefined

$\frac{0}{0}$

Step 2.2

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.

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

Step 2.3

Find the derivative of the numerator and denominator.

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Step 2.3.1

Differentiate the numerator and denominator.

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

Step 2.3.2

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

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

Step 2.3.3

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

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Step 2.3.3.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- \cos (x) \sin (4 x)$ with respect to $x$ is $- \frac{d}{d x} [\cos (x) \sin (4 x)]$ .

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

Step 2.3.3.2

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 $f (x) = \cos (x)$ and $g (x) = \sin (4 x)$ .

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

Step 2.3.3.3

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 $g (x) = 4 x$ .

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Step 2.3.3.3.1

To apply the Chain Rule, set $u_{1}$ as $4 x$ .

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

Step 2.3.3.3.2

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

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

Step 2.3.3.3.3

Replace all occurrences of $u_{1}$ with $4 x$ .

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

Step 2.3.3.4

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

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

Step 2.3.3.5

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

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

Step 2.3.3.6

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

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

Step 2.3.3.7

Multiply $4$ by $1$ .

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

Step 2.3.3.8

Move $4$ to the left of $\cos (4 x)$ .

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

Step 2.3.3.9

Move $4$ to the left of $\cos (x)$ .

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

Step 2.3.4

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

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Step 2.3.4.1

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

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

Step 2.3.4.2

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 $f (x) = \cos (4 x)$ and $g (x) = \sin (x)$ .

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

Step 2.3.4.3

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

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

Step 2.3.4.4

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 $g (x) = 4 x$ .

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Step 2.3.4.4.1

To apply the Chain Rule, set $u_{2}$ as $4 x$ .

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

Step 2.3.4.4.2

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

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

Step 2.3.4.4.3

Replace all occurrences of $u_{2}$ with $4 x$ .

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

Step 2.3.4.5

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

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

Step 2.3.4.6

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

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

Step 2.3.4.7

Multiply $4$ by $1$ .

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

Step 2.3.4.8

Multiply $4$ by $- 1$ .

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

Step 2.3.5

Simplify.

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Step 2.3.5.1

Apply the distributive property.

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

Step 2.3.5.2

Apply the distributive property.

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

Step 2.3.5.3

Combine terms.

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Step 2.3.5.3.1

Multiply $4$ by $- 1$ .

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

Step 2.3.5.3.2

Multiply $- 1$ by $- 1$ .

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

Step 2.3.5.3.3

Multiply $\sin (4 x)$ by $1$ .

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

Step 2.3.5.3.4

Multiply $- 4$ by $- 4$ .

$lim_{x \to 0} \frac{- 4 \cos (x) \cos (4 x) + \sin (4 x) \sin (x) - 4 \cos (4 x) \cos (x) + 16 (\sin (x) (\sin (4 x)))}{\frac{d}{d x} [3 x \cos (3 x) + \sin (3 x)]}$

Step 2.3.5.3.5

Reorder the factors of $- 4 \cos (4 x) \cos (x)$ .

$lim_{x \to 0} \frac{- 4 \cos (x) \cos (4 x) + \sin (4 x) \sin (x) - 4 \cos (x) \cos (4 x) + 16 \sin (x) \sin (4 x)}{\frac{d}{d x} [3 x \cos (3 x) + \sin (3 x)]}$

Step 2.3.5.3.6

Subtract $4 \cos (x) \cos (4 x)$ from $- 4 \cos (x) \cos (4 x)$ .

$lim_{x \to 0} \frac{- 8 \cos (x) \cos (4 x) + \sin (4 x) \sin (x) + 16 \sin (x) \sin (4 x)}{\frac{d}{d x} [3 x \cos (3 x) + \sin (3 x)]}$

Step 2.3.5.3.7

Reorder the factors of $\sin (4 x) \sin (x)$ .

$lim_{x \to 0} \frac{- 8 \cos (x) \cos (4 x) + \sin (x) \sin (4 x) + 16 \sin (x) \sin (4 x)}{\frac{d}{d x} [3 x \cos (3 x) + \sin (3 x)]}$

Step 2.3.5.3.8

Add $\sin (x) \sin (4 x)$ and $16 \sin (x) \sin (4 x)$ .

$lim_{x \to 0} \frac{- 8 \cos (x) \cos (4 x) + 17 \sin (x) \sin (4 x)}{\frac{d}{d x} [3 x \cos (3 x) + \sin (3 x)]}$

Step 2.3.6

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

$lim_{x \to 0} \frac{- 8 \cos (x) \cos (4 x) + 17 \sin (x) \sin (4 x)}{\frac{d}{d x} [3 x \cos (3 x)] + \frac{d}{d x} [\sin (3 x)]}$

Step 2.3.7

Evaluate $\frac{d}{d x} [3 x \cos (3 x)]$ .

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Step 2.3.7.1

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

$lim_{x \to 0} \frac{- 8 \cos (x) \cos (4 x) + 17 \sin (x) \sin (4 x)}{3 \frac{d}{d x} [x \cos (3 x)] + \frac{d}{d x} [\sin (3 x)]}$

Step 2.3.7.2

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 $f (x) = x$ and $g (x) = \cos (3 x)$ .

$lim_{x \to 0} \frac{- 8 \cos (x) \cos (4 x) + 17 \sin (x) \sin (4 x)}{3 (x \frac{d}{d x} [\cos (3 x)] + \cos (3 x) \frac{d}{d x} [x]) + \frac{d}{d x} [\sin (3 x)]}$

Step 2.3.7.3

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 $g (x) = 3 x$ .

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Step 2.3.7.3.1

To apply the Chain Rule, set $u_{1}$ as $3 x$ .

$lim_{x \to 0} \frac{- 8 \cos (x) \cos (4 x) + 17 \sin (x) \sin (4 x)}{3 (x (\frac{d}{d u_{1}} [\cos (u_{1})] \frac{d}{d x} [3 x]) + \cos (3 x) \frac{d}{d x} [x]) + \frac{d}{d x} [\sin (3 x)]}$

Step 2.3.7.3.2

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

$lim_{x \to 0} \frac{- 8 \cos (x) \cos (4 x) + 17 \sin (x) \sin (4 x)}{3 (x (- \sin (u_{1}) \frac{d}{d x} [3 x]) + \cos (3 x) \frac{d}{d x} [x]) + \frac{d}{d x} [\sin (3 x)]}$

Step 2.3.7.3.3

Replace all occurrences of $u_{1}$ with $3 x$ .

$lim_{x \to 0} \frac{- 8 \cos (x) \cos (4 x) + 17 \sin (x) \sin (4 x)}{3 (x (- \sin (3 x) \frac{d}{d x} [3 x]) + \cos (3 x) \frac{d}{d x} [x]) + \frac{d}{d x} [\sin (3 x)]}$

Step 2.3.7.4

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

$lim_{x \to 0} \frac{- 8 \cos (x) \cos (4 x) + 17 \sin (x) \sin (4 x)}{3 (x (- \sin (3 x) (3 \frac{d}{d x} [x])) + \cos (3 x) \frac{d}{d x} [x]) + \frac{d}{d x} [\sin (3 x)]}$

Step 2.3.7.5

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

$lim_{x \to 0} \frac{- 8 \cos (x) \cos (4 x) + 17 \sin (x) \sin (4 x)}{3 (x (- \sin (3 x) (3 \cdot 1)) + \cos (3 x) \frac{d}{d x} [x]) + \frac{d}{d x} [\sin (3 x)]}$

Step 2.3.7.6

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

$lim_{x \to 0} \frac{- 8 \cos (x) \cos (4 x) + 17 \sin (x) \sin (4 x)}{3 (x (- \sin (3 x) (3 \cdot 1)) + \cos (3 x) \cdot 1) + \frac{d}{d x} [\sin (3 x)]}$

Step 2.3.7.7

Multiply $3$ by $1$ .

$lim_{x \to 0} \frac{- 8 \cos (x) \cos (4 x) + 17 \sin (x) \sin (4 x)}{3 (x (- \sin (3 x) \cdot 3) + \cos (3 x) \cdot 1) + \frac{d}{d x} [\sin (3 x)]}$

Step 2.3.7.8

Multiply $3$ by $- 1$ .

$lim_{x \to 0} \frac{- 8 \cos (x) \cos (4 x) + 17 \sin (x) \sin (4 x)}{3 (x (- 3 \sin (3 x)) + \cos (3 x) \cdot 1) + \frac{d}{d x} [\sin (3 x)]}$

Step 2.3.7.9

Multiply $\cos (3 x)$ by $1$ .

$lim_{x \to 0} \frac{- 8 \cos (x) \cos (4 x) + 17 \sin (x) \sin (4 x)}{3 (x (- 3 \sin (3 x)) + \cos (3 x)) + \frac{d}{d x} [\sin (3 x)]}$

Step 2.3.8

Evaluate $\frac{d}{d x} [\sin (3 x)]$ .

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Step 2.3.8.1

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 $g (x) = 3 x$ .

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Step 2.3.8.1.1

To apply the Chain Rule, set $u_{2}$ as $3 x$ .

$lim_{x \to 0} \frac{- 8 \cos (x) \cos (4 x) + 17 \sin (x) \sin (4 x)}{3 (x (- 3 \sin (3 x)) + \cos (3 x)) + \frac{d}{d u_{2}} [\sin (u_{2})] \frac{d}{d x} [3 x]}$

Step 2.3.8.1.2

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

$lim_{x \to 0} \frac{- 8 \cos (x) \cos (4 x) + 17 \sin (x) \sin (4 x)}{3 (x (- 3 \sin (3 x)) + \cos (3 x)) + \cos (u_{2}) \frac{d}{d x} [3 x]}$

Step 2.3.8.1.3

Replace all occurrences of $u_{2}$ with $3 x$ .

$lim_{x \to 0} \frac{- 8 \cos (x) \cos (4 x) + 17 \sin (x) \sin (4 x)}{3 (x (- 3 \sin (3 x)) + \cos (3 x)) + \cos (3 x) \frac{d}{d x} [3 x]}$

Step 2.3.8.2

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

$lim_{x \to 0} \frac{- 8 \cos (x) \cos (4 x) + 17 \sin (x) \sin (4 x)}{3 (x (- 3 \sin (3 x)) + \cos (3 x)) + \cos (3 x) (3 \frac{d}{d x} [x])}$

Step 2.3.8.3

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

$lim_{x \to 0} \frac{- 8 \cos (x) \cos (4 x) + 17 \sin (x) \sin (4 x)}{3 (x (- 3 \sin (3 x)) + \cos (3 x)) + \cos (3 x) (3 \cdot 1)}$

Step 2.3.8.4

Multiply $3$ by $1$ .

$lim_{x \to 0} \frac{- 8 \cos (x) \cos (4 x) + 17 \sin (x) \sin (4 x)}{3 (x (- 3 \sin (3 x)) + \cos (3 x)) + \cos (3 x) \cdot 3}$

Step 2.3.8.5

Move $3$ to the left of $\cos (3 x)$ .

$lim_{x \to 0} \frac{- 8 \cos (x) \cos (4 x) + 17 \sin (x) \sin (4 x)}{3 (x (- 3 \sin (3 x)) + \cos (3 x)) + 3 \cos (3 x)}$

Step 2.3.9

Simplify.

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Step 2.3.9.1

Apply the distributive property.

$lim_{x \to 0} \frac{- 8 \cos (x) \cos (4 x) + 17 \sin (x) \sin (4 x)}{3 (x (- 3 \sin (3 x))) + 3 \cos (3 x) + 3 \cos (3 x)}$

Step 2.3.9.2

Combine terms.

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Step 2.3.9.2.1

Multiply $- 3$ by $3$ .

$lim_{x \to 0} \frac{- 8 \cos (x) \cos (4 x) + 17 \sin (x) \sin (4 x)}{- 9 (x (\sin (3 x))) + 3 \cos (3 x) + 3 \cos (3 x)}$

Step 2.3.9.2.2

Add $3 \cos (3 x)$ and $3 \cos (3 x)$ .

$lim_{x \to 0} \frac{- 8 \cos (x) \cos (4 x) + 17 \sin (x) \sin (4 x)}{- 9 x \sin (3 x) + 6 \cos (3 x)}$

Step 3

Evaluate the limit.

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Step 3.1

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

$\frac{lim_{x \to 0} - 8 \cos (x) \cos (4 x) + 17 \sin (x) \sin (4 x)}{lim_{x \to 0} - 9 x \sin (3 x) + 6 \cos (3 x)}$

Step 3.2

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

$\frac{- lim_{x \to 0} 8 \cos (x) \cos (4 x) + lim_{x \to 0} 17 \sin (x) \sin (4 x)}{lim_{x \to 0} - 9 x \sin (3 x) + 6 \cos (3 x)}$

Step 3.3

Move the term $8$ outside of the limit because it is constant with respect to $x$ .

$\frac{- 8 lim_{x \to 0} \cos (x) \cos (4 x) + lim_{x \to 0} 17 \sin (x) \sin (4 x)}{lim_{x \to 0} - 9 x \sin (3 x) + 6 \cos (3 x)}$

Step 3.4

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

$\frac{- 8 lim_{x \to 0} \cos (x) \cdot lim_{x \to 0} \cos (4 x) + lim_{x \to 0} 17 \sin (x) \sin (4 x)}{lim_{x \to 0} - 9 x \sin (3 x) + 6 \cos (3 x)}$

Step 3.5

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

$\frac{- 8 \cos (lim_{x \to 0} x) \cdot lim_{x \to 0} \cos (4 x) + lim_{x \to 0} 17 \sin (x) \sin (4 x)}{lim_{x \to 0} - 9 x \sin (3 x) + 6 \cos (3 x)}$

Step 3.6

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

$\frac{- 8 \cos (lim_{x \to 0} x) \cdot \cos (lim_{x \to 0} 4 x) + lim_{x \to 0} 17 \sin (x) \sin (4 x)}{lim_{x \to 0} - 9 x \sin (3 x) + 6 \cos (3 x)}$

Step 3.7

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

$\frac{- 8 \cos (lim_{x \to 0} x) \cdot \cos (4 lim_{x \to 0} x) + lim_{x \to 0} 17 \sin (x) \sin (4 x)}{lim_{x \to 0} - 9 x \sin (3 x) + 6 \cos (3 x)}$

Step 3.8

Move the term $17$ outside of the limit because it is constant with respect to $x$ .

$\frac{- 8 \cos (lim_{x \to 0} x) \cdot \cos (4 lim_{x \to 0} x) + 17 lim_{x \to 0} \sin (x) \sin (4 x)}{lim_{x \to 0} - 9 x \sin (3 x) + 6 \cos (3 x)}$

Step 3.9

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

$\frac{- 8 \cos (lim_{x \to 0} x) \cdot \cos (4 lim_{x \to 0} x) + 17 lim_{x \to 0} \sin (x) \cdot lim_{x \to 0} \sin (4 x)}{lim_{x \to 0} - 9 x \sin (3 x) + 6 \cos (3 x)}$

Step 3.10

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

$\frac{- 8 \cos (lim_{x \to 0} x) \cdot \cos (4 lim_{x \to 0} x) + 17 \sin (lim_{x \to 0} x) \cdot lim_{x \to 0} \sin (4 x)}{lim_{x \to 0} - 9 x \sin (3 x) + 6 \cos (3 x)}$

Step 3.11

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

$\frac{- 8 \cos (lim_{x \to 0} x) \cdot \cos (4 lim_{x \to 0} x) + 17 \sin (lim_{x \to 0} x) \cdot \sin (lim_{x \to 0} 4 x)}{lim_{x \to 0} - 9 x \sin (3 x) + 6 \cos (3 x)}$

Step 3.12

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

$\frac{- 8 \cos (lim_{x \to 0} x) \cdot \cos (4 lim_{x \to 0} x) + 17 \sin (lim_{x \to 0} x) \cdot \sin (4 lim_{x \to 0} x)}{lim_{x \to 0} - 9 x \sin (3 x) + 6 \cos (3 x)}$

Step 3.13

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

$\frac{- 8 \cos (lim_{x \to 0} x) \cdot \cos (4 lim_{x \to 0} x) + 17 \sin (lim_{x \to 0} x) \cdot \sin (4 lim_{x \to 0} x)}{- lim_{x \to 0} 9 x \sin (3 x) + lim_{x \to 0} 6 \cos (3 x)}$

Step 3.14

Move the term $9$ outside of the limit because it is constant with respect to $x$ .

$\frac{- 8 \cos (lim_{x \to 0} x) \cdot \cos (4 lim_{x \to 0} x) + 17 \sin (lim_{x \to 0} x) \cdot \sin (4 lim_{x \to 0} x)}{- 9 lim_{x \to 0} x \sin (3 x) + lim_{x \to 0} 6 \cos (3 x)}$

Step 3.15

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

$\frac{- 8 \cos (lim_{x \to 0} x) \cdot \cos (4 lim_{x \to 0} x) + 17 \sin (lim_{x \to 0} x) \cdot \sin (4 lim_{x \to 0} x)}{- 9 lim_{x \to 0} x \cdot lim_{x \to 0} \sin (3 x) + lim_{x \to 0} 6 \cos (3 x)}$

Step 3.16

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

$\frac{- 8 \cos (lim_{x \to 0} x) \cdot \cos (4 lim_{x \to 0} x) + 17 \sin (lim_{x \to 0} x) \cdot \sin (4 lim_{x \to 0} x)}{- 9 lim_{x \to 0} x \cdot \sin (lim_{x \to 0} 3 x) + lim_{x \to 0} 6 \cos (3 x)}$

Step 3.17

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

$\frac{- 8 \cos (lim_{x \to 0} x) \cdot \cos (4 lim_{x \to 0} x) + 17 \sin (lim_{x \to 0} x) \cdot \sin (4 lim_{x \to 0} x)}{- 9 lim_{x \to 0} x \cdot \sin (3 lim_{x \to 0} x) + lim_{x \to 0} 6 \cos (3 x)}$

Step 3.18

Move the term $6$ outside of the limit because it is constant with respect to $x$ .

$\frac{- 8 \cos (lim_{x \to 0} x) \cdot \cos (4 lim_{x \to 0} x) + 17 \sin (lim_{x \to 0} x) \cdot \sin (4 lim_{x \to 0} x)}{- 9 lim_{x \to 0} x \cdot \sin (3 lim_{x \to 0} x) + 6 lim_{x \to 0} \cos (3 x)}$

Step 3.19

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

$\frac{- 8 \cos (lim_{x \to 0} x) \cdot \cos (4 lim_{x \to 0} x) + 17 \sin (lim_{x \to 0} x) \cdot \sin (4 lim_{x \to 0} x)}{- 9 lim_{x \to 0} x \cdot \sin (3 lim_{x \to 0} x) + 6 \cos (lim_{x \to 0} 3 x)}$

Step 3.20

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

$\frac{- 8 \cos (lim_{x \to 0} x) \cdot \cos (4 lim_{x \to 0} x) + 17 \sin (lim_{x \to 0} x) \cdot \sin (4 lim_{x \to 0} x)}{- 9 lim_{x \to 0} x \cdot \sin (3 lim_{x \to 0} x) + 6 \cos (3 lim_{x \to 0} x)}$

Step 4

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

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Step 4.1

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

$\frac{- 8 \cos (0) \cdot \cos (4 lim_{x \to 0} x) + 17 \sin (lim_{x \to 0} x) \cdot \sin (4 lim_{x \to 0} x)}{- 9 lim_{x \to 0} x \cdot \sin (3 lim_{x \to 0} x) + 6 \cos (3 lim_{x \to 0} x)}$

Step 4.2

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

$\frac{- 8 \cos (0) \cdot \cos (4 \cdot 0) + 17 \sin (lim_{x \to 0} x) \cdot \sin (4 lim_{x \to 0} x)}{- 9 lim_{x \to 0} x \cdot \sin (3 lim_{x \to 0} x) + 6 \cos (3 lim_{x \to 0} x)}$

Step 4.3

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

$\frac{- 8 \cos (0) \cdot \cos (4 \cdot 0) + 17 \sin (0) \cdot \sin (4 lim_{x \to 0} x)}{- 9 lim_{x \to 0} x \cdot \sin (3 lim_{x \to 0} x) + 6 \cos (3 lim_{x \to 0} x)}$

Step 4.4

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

$\frac{- 8 \cos (0) \cdot \cos (4 \cdot 0) + 17 \sin (0) \cdot \sin (4 \cdot 0)}{- 9 lim_{x \to 0} x \cdot \sin (3 lim_{x \to 0} x) + 6 \cos (3 lim_{x \to 0} x)}$

Step 4.5

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

$\frac{- 8 \cos (0) \cdot \cos (4 \cdot 0) + 17 \sin (0) \cdot \sin (4 \cdot 0)}{- 9 \cdot 0 \cdot \sin (3 lim_{x \to 0} x) + 6 \cos (3 lim_{x \to 0} x)}$

Step 4.6

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

$\frac{- 8 \cos (0) \cdot \cos (4 \cdot 0) + 17 \sin (0) \cdot \sin (4 \cdot 0)}{- 9 \cdot 0 \cdot \sin (3 \cdot 0) + 6 \cos (3 lim_{x \to 0} x)}$

Step 4.7

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

$\frac{- 8 \cos (0) \cdot \cos (4 \cdot 0) + 17 \sin (0) \cdot \sin (4 \cdot 0)}{- 9 \cdot 0 \cdot \sin (3 \cdot 0) + 6 \cos (3 \cdot 0)}$

Step 5

Simplify the answer.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

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

$\frac{- 8 \cdot 1 \cdot \cos (4 \cdot 0) + 17 \sin (0) \cdot \sin (4 \cdot 0)}{- 9 \cdot 0 \cdot \sin (3 \cdot 0) + 6 \cos (3 \cdot 0)}$

Step 5.1.2

Multiply $- 8$ by $1$ .

$\frac{- 8 \cdot \cos (4 \cdot 0) + 17 \sin (0) \cdot \sin (4 \cdot 0)}{- 9 \cdot 0 \cdot \sin (3 \cdot 0) + 6 \cos (3 \cdot 0)}$

Step 5.1.3

Multiply $4$ by $0$ .

$\frac{- 8 \cdot \cos (0) + 17 \sin (0) \cdot \sin (4 \cdot 0)}{- 9 \cdot 0 \cdot \sin (3 \cdot 0) + 6 \cos (3 \cdot 0)}$

Step 5.1.4

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

$\frac{- 8 \cdot 1 + 17 \sin (0) \cdot \sin (4 \cdot 0)}{- 9 \cdot 0 \cdot \sin (3 \cdot 0) + 6 \cos (3 \cdot 0)}$

Step 5.1.5

Multiply $- 8$ by $1$ .

$\frac{- 8 + 17 \sin (0) \cdot \sin (4 \cdot 0)}{- 9 \cdot 0 \cdot \sin (3 \cdot 0) + 6 \cos (3 \cdot 0)}$

Step 5.1.6

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

$\frac{- 8 + 17 \cdot 0 \cdot \sin (4 \cdot 0)}{- 9 \cdot 0 \cdot \sin (3 \cdot 0) + 6 \cos (3 \cdot 0)}$

Step 5.1.7

Multiply $17$ by $0$ .

$\frac{- 8 + 0 \cdot \sin (4 \cdot 0)}{- 9 \cdot 0 \cdot \sin (3 \cdot 0) + 6 \cos (3 \cdot 0)}$

Step 5.1.8

Multiply $4$ by $0$ .

$\frac{- 8 + 0 \cdot \sin (0)}{- 9 \cdot 0 \cdot \sin (3 \cdot 0) + 6 \cos (3 \cdot 0)}$

Step 5.1.9

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

$\frac{- 8 + 0 \cdot 0}{- 9 \cdot 0 \cdot \sin (3 \cdot 0) + 6 \cos (3 \cdot 0)}$

Step 5.1.10

Multiply $0$ by $0$ .

$\frac{- 8 + 0}{- 9 \cdot 0 \cdot \sin (3 \cdot 0) + 6 \cos (3 \cdot 0)}$

Step 5.1.11

Add $- 8$ and $0$ .

$\frac{- 8}{- 9 \cdot 0 \cdot \sin (3 \cdot 0) + 6 \cos (3 \cdot 0)}$

Step 5.2

Simplify the denominator.

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Step 5.2.1

Multiply $- 9$ by $0$ .

$\frac{- 8}{0 \cdot \sin (3 \cdot 0) + 6 \cos (3 \cdot 0)}$

Step 5.2.2

Multiply $3$ by $0$ .

$\frac{- 8}{0 \cdot \sin (0) + 6 \cos (3 \cdot 0)}$

Step 5.2.3

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

$\frac{- 8}{0 \cdot 0 + 6 \cos (3 \cdot 0)}$

Step 5.2.4

Multiply $0$ by $0$ .

$\frac{- 8}{0 + 6 \cos (3 \cdot 0)}$

Step 5.2.5

Multiply $3$ by $0$ .

$\frac{- 8}{0 + 6 \cos (0)}$

Step 5.2.6

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

$\frac{- 8}{0 + 6 \cdot 1}$

Step 5.2.7

Multiply $6$ by $1$ .

$\frac{- 8}{0 + 6}$

Step 5.2.8

Add $0$ and $6$ .

$\frac{- 8}{6}$

Step 5.3

Cancel the common factor of $- 8$ and $6$ .

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Step 5.3.1

Factor $2$ out of $- 8$ .

$\frac{2 (- 4)}{6}$

Step 5.3.2

Cancel the common factors.

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Step 5.3.2.1

Factor $2$ out of $6$ .

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

Step 5.3.2.2

Cancel the common factor.

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

Step 5.3.2.3

Rewrite the expression.

$\frac{- 4}{3}$

Step 5.4

Move the negative in front of the fraction.

$- \frac{4}{3}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$- \frac{4}{3}$

Decimal Form:

$- 1.\overline{3}$