Calculus Examples

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

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.

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

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 $- 2$ .

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

Step 1.2.2

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

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

Step 1.2.3

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

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

Step 1.2.4

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $- 2$ .

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

Step 1.2.5

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

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

Step 1.2.6

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

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

Step 1.2.7

Evaluate the limits by plugging in $- 2$ for all occurrences of $x$ .

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Step 1.2.7.1

Evaluate the limit of $x$ by plugging in $- 2$ for $x$ .

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

Step 1.2.7.2

Evaluate the limit of $x$ by plugging in $- 2$ for $x$ .

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

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 $- 2$ and $2$ .

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

Step 1.2.8.1.2

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

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

Step 1.2.8.1.3

Multiply $4$ by $1$ .

$\frac{4 - {(- 2)}^{2}}{lim_{x \to - 2} 4 e^{- 4 - 2 x} - x^{2}}$

Step 1.2.8.1.4

Raise $- 2$ to the power of $2$ .

$\frac{4 - 1 \cdot 4}{lim_{x \to - 2} 4 e^{- 4 - 2 x} - x^{2}}$

Step 1.2.8.1.5

Multiply $- 1$ by $4$ .

$\frac{4 - 4}{lim_{x \to - 2} 4 e^{- 4 - 2 x} - x^{2}}$

Step 1.2.8.2

Subtract $4$ from $4$ .

$\frac{0}{lim_{x \to - 2} 4 e^{- 4 - 2 x} - x^{2}}$

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 $- 2$ .

$\frac{0}{lim_{x \to - 2} 4 e^{- 4 - 2 x} - lim_{x \to - 2} x^{2}}$

Step 1.3.2

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

$\frac{0}{4 lim_{x \to - 2} e^{- 4 - 2 x} - lim_{x \to - 2} x^{2}}$

Step 1.3.3

Move the limit into the exponent.

$\frac{0}{4 e^{lim_{x \to - 2} - 4 - 2 x} - lim_{x \to - 2} x^{2}}$

Step 1.3.4

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $- 2$ .

$\frac{0}{4 e^{- lim_{x \to - 2} 4 - lim_{x \to - 2} 2 x} - lim_{x \to - 2} x^{2}}$

Step 1.3.5

Evaluate the limit of $4$ which is constant as $x$ approaches $- 2$ .

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

Step 1.3.6

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

$\frac{0}{4 e^{- 4 - 2 lim_{x \to - 2} x} - lim_{x \to - 2} x^{2}}$

Step 1.3.7

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

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

Step 1.3.8

Evaluate the limits by plugging in $- 2$ for all occurrences of $x$ .

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Step 1.3.8.1

Evaluate the limit of $x$ by plugging in $- 2$ for $x$ .

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

Step 1.3.8.2

Evaluate the limit of $x$ by plugging in $- 2$ for $x$ .

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

Step 1.3.9

Simplify the answer.

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Step 1.3.9.1

Simplify each term.

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Step 1.3.9.1.1

Multiply $- 2$ by $- 2$ .

$\frac{0}{4 e^{- 4 + 4} - {(- 2)}^{2}}$

Step 1.3.9.1.2

Add $- 4$ and $4$ .

$\frac{0}{4 e^{0} - {(- 2)}^{2}}$

Step 1.3.9.1.3

Anything raised to $0$ is $1$ .

$\frac{0}{4 \cdot 1 - {(- 2)}^{2}}$

Step 1.3.9.1.4

Multiply $4$ by $1$ .

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

Step 1.3.9.1.5

Raise $- 2$ to the power of $2$ .

$\frac{0}{4 - 1 \cdot 4}$

Step 1.3.9.1.6

Multiply $- 1$ by $4$ .

$\frac{0}{4 - 4}$

Step 1.3.9.2

Subtract $4$ from $4$ .

$\frac{0}{0}$

Step 1.3.9.3

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

Undefined

$\frac{0}{0}$

Step 1.3.10

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.

$lim_{x \to - 2} \frac{4 \cos (x + 2) - x^{2}}{4 e^{- 4 - 2 x} - x^{2}} = lim_{x \to - 2} \frac{\frac{d}{d x} [4 \cos (x + 2) - x^{2}]}{\frac{d}{d x} [4 e^{- 4 - 2 x} - x^{2}]}$

Step 3

Find the derivative of the numerator and denominator.

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Step 3.1

Differentiate the numerator and denominator.

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

Step 3.2

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

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

Step 3.3

Evaluate $\frac{d}{d x} [4 \cos (x + 2)]$ .

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Step 3.3.1

Since $4$ is constant with respect to $x$ , the derivative of $4 \cos (x + 2)$ with respect to $x$ is $4 \frac{d}{d x} [\cos (x + 2)]$ .

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

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 $f (x) = \cos (x)$ and $g (x) = x + 2$ .

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Step 3.3.2.1

To apply the Chain Rule, set $u$ as $x + 2$ .

$lim_{x \to - 2} \frac{4 (\frac{d}{d u} [\cos (u)] \frac{d}{d x} [x + 2]) + \frac{d}{d x} [- x^{2}]}{\frac{d}{d x} [4 e^{- 4 - 2 x} - x^{2}]}$

Step 3.3.2.2

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

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

Step 3.3.2.3

Replace all occurrences of $u$ with $x + 2$ .

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

Step 3.3.3

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

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

Step 3.3.4

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

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

Step 3.3.5

Since $2$ is constant with respect to $x$ , the derivative of $2$ with respect to $x$ is $0$ .

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

Step 3.3.6

Add $1$ and $0$ .

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

Step 3.3.7

Multiply $- 1$ by $1$ .

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

Step 3.3.8

Multiply $- 1$ by $4$ .

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

Step 3.4

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

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Step 3.4.1

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

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

Step 3.4.2

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

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

Step 3.4.3

Multiply $2$ by $- 1$ .

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

Step 3.5

Reorder terms.

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

Step 3.6

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

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

Step 3.7

Evaluate $\frac{d}{d x} [4 e^{- 4 - 2 x}]$ .

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Step 3.7.1

Since $4$ is constant with respect to $x$ , the derivative of $4 e^{- 4 - 2 x}$ with respect to $x$ is $4 \frac{d}{d x} [e^{- 4 - 2 x}]$ .

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

Step 3.7.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = e^{x}$ and $g (x) = - 4 - 2 x$ .

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Step 3.7.2.1

To apply the Chain Rule, set $u$ as $- 4 - 2 x$ .

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

Step 3.7.2.2

Differentiate using the Exponential Rule which states that $\frac{d}{d u} [a^{u}]$ is $a^{u} \ln (a)$ where $a$ = $e$ .

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

Step 3.7.2.3

Replace all occurrences of $u$ with $- 4 - 2 x$ .

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

Step 3.7.3

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

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

Step 3.7.4

Since $- 4$ is constant with respect to $x$ , the derivative of $- 4$ with respect to $x$ is $0$ .

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

Step 3.7.5

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

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

Step 3.7.6

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

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

Step 3.7.7

Multiply $- 2$ by $1$ .

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

Step 3.7.8

Subtract $2$ from $0$ .

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

Step 3.7.9

Move $- 2$ to the left of $e^{- 4 - 2 x}$ .

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

Step 3.7.10

Multiply $- 2$ by $4$ .

$lim_{x \to - 2} \frac{- 2 x - 4 \sin (x + 2)}{- 8 e^{- 4 - 2 x} + \frac{d}{d x} [- x^{2}]}$

Step 3.8

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

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Step 3.8.1

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

$lim_{x \to - 2} \frac{- 2 x - 4 \sin (x + 2)}{- 8 e^{- 4 - 2 x} - \frac{d}{d x} [x^{2}]}$

Step 3.8.2

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

$lim_{x \to - 2} \frac{- 2 x - 4 \sin (x + 2)}{- 8 e^{- 4 - 2 x} - (2 x)}$

Step 3.8.3

Multiply $2$ by $- 1$ .

$lim_{x \to - 2} \frac{- 2 x - 4 \sin (x + 2)}{- 8 e^{- 4 - 2 x} - 2 x}$

Step 3.9

Reorder terms.

$lim_{x \to - 2} \frac{- 2 x - 4 \sin (x + 2)}{- 2 x - 8 e^{- 4 - 2 x}}$

Step 4

Cancel the common factor of $- 2 x - 4 \sin (x + 2)$ and $- 2 x - 8 e^{- 4 - 2 x}$ .

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Step 4.1

Factor $2$ out of $- 2 x$ .

$lim_{x \to - 2} \frac{2 (- x) - 4 \sin (x + 2)}{- 2 x - 8 e^{- 4 - 2 x}}$

Step 4.2

Factor $2$ out of $- 4 \sin (x + 2)$ .

$lim_{x \to - 2} \frac{2 (- x) + 2 (- 2 \sin (x + 2))}{- 2 x - 8 e^{- 4 - 2 x}}$

Step 4.3

Factor $2$ out of $2 (- x) + 2 (- 2 \sin (x + 2))$ .

$lim_{x \to - 2} \frac{2 (- x - 2 \sin (x + 2))}{- 2 x - 8 e^{- 4 - 2 x}}$

Step 4.4

Cancel the common factors.

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Step 4.4.1

Factor $2$ out of $- 2 x$ .

$lim_{x \to - 2} \frac{2 (- x - 2 \sin (x + 2))}{2 (- x) - 8 e^{- 4 - 2 x}}$

Step 4.4.2

Factor $2$ out of $- 8 e^{- 4 - 2 x}$ .

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

Step 4.4.3

Factor $2$ out of $2 (- x) + 2 (- 4 e^{- 4 - 2 x})$ .

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

Step 4.4.4

Cancel the common factor.

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

Step 4.4.5

Rewrite the expression.

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

Step 5

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $- 2$ .

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

Step 6

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $- 2$ .

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

Step 7

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

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

Step 8

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

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

Step 9

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $- 2$ .

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

Step 10

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

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

Step 11

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $- 2$ .

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

Step 12

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

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

Step 13

Move the limit into the exponent.

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

Step 14

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $- 2$ .

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

Step 15

Evaluate the limit of $4$ which is constant as $x$ approaches $- 2$ .

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

Step 16

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

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

Step 17

Evaluate the limits by plugging in $- 2$ for all occurrences of $x$ .

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Step 17.1

Evaluate the limit of $x$ by plugging in $- 2$ for $x$ .

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

Step 17.2

Evaluate the limit of $x$ by plugging in $- 2$ for $x$ .

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

Step 17.3

Evaluate the limit of $x$ by plugging in $- 2$ for $x$ .

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

Step 17.4

Evaluate the limit of $x$ by plugging in $- 2$ for $x$ .

$\frac{2 - 2 \sin (- 2 + 2)}{2 - 4 e^{- 4 - 2 \cdot - 2}}$

Step 18

Simplify the answer.

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Step 18.1

Cancel the common factor of $2 - 2 \sin (- 2 + 2)$ and $2 - 4 e^{- 4 - 2 \cdot - 2}$ .

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Step 18.1.1

Factor $2$ out of $2$ .

$\frac{2 \cdot 1 - 2 \sin (- 2 + 2)}{2 - 4 e^{- 4 - 2 \cdot - 2}}$

Step 18.1.2

Factor $2$ out of $- 2 \sin (- 2 + 2)$ .

$\frac{2 \cdot 1 + 2 (- \sin (- 2 + 2))}{2 - 4 e^{- 4 - 2 \cdot - 2}}$

Step 18.1.3

Factor $2$ out of $2 (1) + 2 (- \sin (- 2 + 2))$ .

$\frac{2 (1 - \sin (- 2 + 2))}{2 - 4 e^{- 4 - 2 \cdot - 2}}$

Step 18.1.4

Cancel the common factors.

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Step 18.1.4.1

Factor $2$ out of $2$ .

$\frac{2 (1 - \sin (- 2 + 2))}{2 (1) - 4 e^{- 4 - 2 \cdot - 2}}$

Step 18.1.4.2

Factor $2$ out of $- 4 e^{- 4 - 2 \cdot - 2}$ .

$\frac{2 (1 - \sin (- 2 + 2))}{2 (1) + 2 (- 2 e^{- 4 - 2 \cdot - 2})}$

Step 18.1.4.3

Factor $2$ out of $2 (1) + 2 (- 2 e^{- 4 - 2 \cdot - 2})$ .

$\frac{2 (1 - \sin (- 2 + 2))}{2 (1 - 2 e^{- 4 - 2 \cdot - 2})}$

Step 18.1.4.4

Cancel the common factor.

$\frac{2 (1 - \sin (- 2 + 2))}{2 (1 - 2 e^{- 4 - 2 \cdot - 2})}$

Step 18.1.4.5

Rewrite the expression.

$\frac{1 - \sin (- 2 + 2)}{1 - 2 e^{- 4 - 2 \cdot - 2}}$

Step 18.2

Simplify the numerator.

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Step 18.2.1

Add $- 2$ and $2$ .

$\frac{1 - \sin (0)}{1 - 2 e^{- 4 - 2 \cdot - 2}}$

Step 18.2.2

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

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

Step 18.2.3

Multiply $- 1$ by $0$ .

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

Step 18.2.4

Add $1$ and $0$ .

$\frac{1}{1 - 2 e^{- 4 - 2 \cdot - 2}}$

Step 18.3

Simplify the denominator.

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Step 18.3.1

Multiply $- 2$ by $- 2$ .

$\frac{1}{1 - 2 e^{- 4 + 4}}$

Step 18.3.2

Add $- 4$ and $4$ .

$\frac{1}{1 - 2 e^{0}}$

Step 18.3.3

Anything raised to $0$ is $1$ .

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

Step 18.3.4

Multiply $- 2$ by $1$ .

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

Step 18.3.5

Subtract $2$ from $1$ .

$\frac{1}{- 1}$

Step 18.4

Divide $1$ by $- 1$ .

$- 1$