Calculus Examples

$lim_{x \to 0} \frac{(\cos (x)) (\cos (x)) + 6 \cos (x) - 7}{\cos (x) - 1}$

Step 1

Combine factors.

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Step 1.1

Raise $\cos (x)$ to the power of $1$ .

$lim_{x \to 0} \frac{\cos^{1} (x) \cos (x) + 6 \cos (x) - 7}{\cos (x) - 1}$

Step 1.2

Raise $\cos (x)$ to the power of $1$ .

$lim_{x \to 0} \frac{\cos^{1} (x) \cos^{1} (x) + 6 \cos (x) - 7}{\cos (x) - 1}$

Step 1.3

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

$lim_{x \to 0} \frac{{\cos (x)}^{1 + 1} + 6 \cos (x) - 7}{\cos (x) - 1}$

Step 1.4

Add $1$ and $1$ .

$lim_{x \to 0} \frac{\cos^{2} (x) + 6 \cos (x) - 7}{\cos (x) - 1}$

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.

$\frac{lim_{x \to 0} \cos^{2} (x) + 6 \cos (x) - 7}{lim_{x \to 0} \cos (x) - 1}$

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 $0$ .

$\frac{lim_{x \to 0} \cos^{2} (x) + lim_{x \to 0} 6 \cos (x) - lim_{x \to 0} 7}{lim_{x \to 0} \cos (x) - 1}$

Step 2.1.2.2

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

$\frac{{(lim_{x \to 0} \cos (x))}^{2} + lim_{x \to 0} 6 \cos (x) - lim_{x \to 0} 7}{lim_{x \to 0} \cos (x) - 1}$

Step 2.1.2.3

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

$\frac{\cos^{2} (lim_{x \to 0} x) + lim_{x \to 0} 6 \cos (x) - lim_{x \to 0} 7}{lim_{x \to 0} \cos (x) - 1}$

Step 2.1.2.4

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

$\frac{\cos^{2} (lim_{x \to 0} x) + 6 lim_{x \to 0} \cos (x) - lim_{x \to 0} 7}{lim_{x \to 0} \cos (x) - 1}$

Step 2.1.2.5

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

$\frac{\cos^{2} (lim_{x \to 0} x) + 6 \cos (lim_{x \to 0} x) - lim_{x \to 0} 7}{lim_{x \to 0} \cos (x) - 1}$

Step 2.1.2.6

Evaluate the limit of $7$ which is constant as $x$ approaches $0$ .

$\frac{\cos^{2} (lim_{x \to 0} x) + 6 \cos (lim_{x \to 0} x) - 1 \cdot 7}{lim_{x \to 0} \cos (x) - 1}$

Step 2.1.2.7

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

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Step 2.1.2.7.1

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

$\frac{\cos^{2} (0) + 6 \cos (lim_{x \to 0} x) - 1 \cdot 7}{lim_{x \to 0} \cos (x) - 1}$

Step 2.1.2.7.2

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

$\frac{\cos^{2} (0) + 6 \cos (0) - 1 \cdot 7}{lim_{x \to 0} \cos (x) - 1}$

Step 2.1.2.8

Simplify the answer.

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Step 2.1.2.8.1

Simplify each term.

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Step 2.1.2.8.1.1

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

$\frac{1^{2} + 6 \cos (0) - 1 \cdot 7}{lim_{x \to 0} \cos (x) - 1}$

Step 2.1.2.8.1.2

One to any power is one.

$\frac{1 + 6 \cos (0) - 1 \cdot 7}{lim_{x \to 0} \cos (x) - 1}$

Step 2.1.2.8.1.3

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

$\frac{1 + 6 \cdot 1 - 1 \cdot 7}{lim_{x \to 0} \cos (x) - 1}$

Step 2.1.2.8.1.4

Multiply $6$ by $1$ .

$\frac{1 + 6 - 1 \cdot 7}{lim_{x \to 0} \cos (x) - 1}$

Step 2.1.2.8.1.5

Multiply $- 1$ by $7$ .

$\frac{1 + 6 - 7}{lim_{x \to 0} \cos (x) - 1}$

Step 2.1.2.8.2

Add $1$ and $6$ .

$\frac{7 - 7}{lim_{x \to 0} \cos (x) - 1}$

Step 2.1.2.8.3

Subtract $7$ from $7$ .

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

Step 2.1.3

Evaluate the limit of the denominator.

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Step 2.1.3.1

Evaluate the limit.

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Step 2.1.3.1.1

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

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

Step 2.1.3.1.2

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

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

Step 2.1.3.1.3

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

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

Step 2.1.3.2

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

$\frac{0}{\cos (0) - 1 \cdot 1}$

Step 2.1.3.3

Simplify the answer.

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Step 2.1.3.3.1

Simplify each term.

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Step 2.1.3.3.1.1

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

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

Step 2.1.3.3.1.2

Multiply $- 1$ by $1$ .

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

Step 2.1.3.3.2

Subtract $1$ from $1$ .

$\frac{0}{0}$

Step 2.1.3.3.3

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

Undefined

$\frac{0}{0}$

Step 2.1.3.4

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.

$lim_{x \to 0} \frac{\cos^{2} (x) + 6 \cos (x) - 7}{\cos (x) - 1} = lim_{x \to 0} \frac{\frac{d}{d x} [\cos^{2} (x) + 6 \cos (x) - 7]}{\frac{d}{d x} [\cos (x) - 1]}$

Step 2.3

Find the derivative of the numerator and denominator.

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Step 2.3.1

Differentiate the numerator and denominator.

$lim_{x \to 0} \frac{\frac{d}{d x} [\cos^{2} (x) + 6 \cos (x) - 7]}{\frac{d}{d x} [\cos (x) - 1]}$

Step 2.3.2

By the Sum Rule, the derivative of $\cos^{2} (x) + 6 \cos (x) - 7$ with respect to $x$ is $\frac{d}{d x} [\cos^{2} (x)] + \frac{d}{d x} [6 \cos (x)] + \frac{d}{d x} [- 7]$ .

$lim_{x \to 0} \frac{\frac{d}{d x} [\cos^{2} (x)] + \frac{d}{d x} [6 \cos (x)] + \frac{d}{d x} [- 7]}{\frac{d}{d x} [\cos (x) - 1]}$

Step 2.3.3

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

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Step 2.3.3.1

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

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Step 2.3.3.1.1

To apply the Chain Rule, set $u$ as $\cos (x)$ .

$lim_{x \to 0} \frac{\frac{d}{d u} [u^{2}] \frac{d}{d x} [\cos (x)] + \frac{d}{d x} [6 \cos (x)] + \frac{d}{d x} [- 7]}{\frac{d}{d x} [\cos (x) - 1]}$

Step 2.3.3.1.2

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

$lim_{x \to 0} \frac{2 u \frac{d}{d x} [\cos (x)] + \frac{d}{d x} [6 \cos (x)] + \frac{d}{d x} [- 7]}{\frac{d}{d x} [\cos (x) - 1]}$

Step 2.3.3.1.3

Replace all occurrences of $u$ with $\cos (x)$ .

$lim_{x \to 0} \frac{2 \cos (x) \frac{d}{d x} [\cos (x)] + \frac{d}{d x} [6 \cos (x)] + \frac{d}{d x} [- 7]}{\frac{d}{d x} [\cos (x) - 1]}$

Step 2.3.3.2

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

$lim_{x \to 0} \frac{2 \cos (x) (- \sin (x)) + \frac{d}{d x} [6 \cos (x)] + \frac{d}{d x} [- 7]}{\frac{d}{d x} [\cos (x) - 1]}$

Step 2.3.3.3

Multiply $- 1$ by $2$ .

$lim_{x \to 0} \frac{- 2 \cos (x) \sin (x) + \frac{d}{d x} [6 \cos (x)] + \frac{d}{d x} [- 7]}{\frac{d}{d x} [\cos (x) - 1]}$

Step 2.3.4

Evaluate $\frac{d}{d x} [6 \cos (x)]$ .

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Step 2.3.4.1

Since $6$ is constant with respect to $x$ , the derivative of $6 \cos (x)$ with respect to $x$ is $6 \frac{d}{d x} [\cos (x)]$ .

$lim_{x \to 0} \frac{- 2 \cos (x) \sin (x) + 6 \frac{d}{d x} [\cos (x)] + \frac{d}{d x} [- 7]}{\frac{d}{d x} [\cos (x) - 1]}$

Step 2.3.4.2

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

$lim_{x \to 0} \frac{- 2 \cos (x) \sin (x) + 6 (- \sin (x)) + \frac{d}{d x} [- 7]}{\frac{d}{d x} [\cos (x) - 1]}$

Step 2.3.4.3

Multiply $- 1$ by $6$ .

$lim_{x \to 0} \frac{- 2 \cos (x) \sin (x) - 6 \sin (x) + \frac{d}{d x} [- 7]}{\frac{d}{d x} [\cos (x) - 1]}$

Step 2.3.5

Since $- 7$ is constant with respect to $x$ , the derivative of $- 7$ with respect to $x$ is $0$ .

$lim_{x \to 0} \frac{- 2 \cos (x) \sin (x) - 6 \sin (x) + 0}{\frac{d}{d x} [\cos (x) - 1]}$

Step 2.3.6

Add $- 2 \cos (x) \sin (x) - 6 \sin (x)$ and $0$ .

$lim_{x \to 0} \frac{- 2 \cos (x) \sin (x) - 6 \sin (x)}{\frac{d}{d x} [\cos (x) - 1]}$

Step 2.3.7

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

$lim_{x \to 0} \frac{- 2 \cos (x) \sin (x) - 6 \sin (x)}{\frac{d}{d x} [\cos (x)] + \frac{d}{d x} [- 1]}$

Step 2.3.8

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

$lim_{x \to 0} \frac{- 2 \cos (x) \sin (x) - 6 \sin (x)}{- \sin (x) + \frac{d}{d x} [- 1]}$

Step 2.3.9

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

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

Step 2.3.10

Add $- \sin (x)$ and $0$ .

$lim_{x \to 0} \frac{- 2 \cos (x) \sin (x) - 6 \sin (x)}{- \sin (x)}$

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.

$\frac{lim_{x \to 0} - 2 \cos (x) \sin (x) - 6 \sin (x)}{lim_{x \to 0} - \sin (x)}$

Step 3.1.2

Evaluate the limit of the numerator.

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Step 3.1.2.1

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

$\frac{- lim_{x \to 0} 2 \cos (x) \sin (x) - lim_{x \to 0} 6 \sin (x)}{lim_{x \to 0} - \sin (x)}$

Step 3.1.2.2

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

$\frac{- 2 lim_{x \to 0} \cos (x) \sin (x) - lim_{x \to 0} 6 \sin (x)}{lim_{x \to 0} - \sin (x)}$

Step 3.1.2.3

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

$\frac{- 2 lim_{x \to 0} \cos (x) \cdot lim_{x \to 0} \sin (x) - lim_{x \to 0} 6 \sin (x)}{lim_{x \to 0} - \sin (x)}$

Step 3.1.2.4

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

$\frac{- 2 \cos (lim_{x \to 0} x) \cdot lim_{x \to 0} \sin (x) - lim_{x \to 0} 6 \sin (x)}{lim_{x \to 0} - \sin (x)}$

Step 3.1.2.5

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

$\frac{- 2 \cos (lim_{x \to 0} x) \cdot \sin (lim_{x \to 0} x) - lim_{x \to 0} 6 \sin (x)}{lim_{x \to 0} - \sin (x)}$

Step 3.1.2.6

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

$\frac{- 2 \cos (lim_{x \to 0} x) \cdot \sin (lim_{x \to 0} x) - 6 lim_{x \to 0} \sin (x)}{lim_{x \to 0} - \sin (x)}$

Step 3.1.2.7

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

$\frac{- 2 \cos (lim_{x \to 0} x) \cdot \sin (lim_{x \to 0} x) - 6 \sin (lim_{x \to 0} x)}{lim_{x \to 0} - \sin (x)}$

Step 3.1.2.8

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

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Step 3.1.2.8.1

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

$\frac{- 2 \cos (0) \cdot \sin (lim_{x \to 0} x) - 6 \sin (lim_{x \to 0} x)}{lim_{x \to 0} - \sin (x)}$

Step 3.1.2.8.2

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

$\frac{- 2 \cos (0) \cdot \sin (0) - 6 \sin (lim_{x \to 0} x)}{lim_{x \to 0} - \sin (x)}$

Step 3.1.2.8.3

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

$\frac{- 2 \cos (0) \cdot \sin (0) - 6 \sin (0)}{lim_{x \to 0} - \sin (x)}$

Step 3.1.2.9

Simplify the answer.

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Step 3.1.2.9.1

Simplify each term.

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Step 3.1.2.9.1.1

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

$\frac{- 2 \cdot 1 \cdot \sin (0) - 6 \sin (0)}{lim_{x \to 0} - \sin (x)}$

Step 3.1.2.9.1.2

Multiply $- 2$ by $1$ .

$\frac{- 2 \cdot \sin (0) - 6 \sin (0)}{lim_{x \to 0} - \sin (x)}$

Step 3.1.2.9.1.3

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

$\frac{- 2 \cdot 0 - 6 \sin (0)}{lim_{x \to 0} - \sin (x)}$

Step 3.1.2.9.1.4

Multiply $- 2$ by $0$ .

$\frac{0 - 6 \sin (0)}{lim_{x \to 0} - \sin (x)}$

Step 3.1.2.9.1.5

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

$\frac{0 - 6 \cdot 0}{lim_{x \to 0} - \sin (x)}$

Step 3.1.2.9.1.6

Multiply $- 6$ by $0$ .

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

Step 3.1.2.9.2

Add $0$ and $0$ .

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

Step 3.1.3

Evaluate the limit of the denominator.

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Step 3.1.3.1

Evaluate the limit.

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Step 3.1.3.1.1

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

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

Step 3.1.3.1.2

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

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

Step 3.1.3.2

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

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

Step 3.1.3.3

Simplify the answer.

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Step 3.1.3.3.1

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

$\frac{0}{- 0}$

Step 3.1.3.3.2

Multiply $- 1$ by $0$ .

$\frac{0}{0}$

Step 3.1.3.3.3

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

Undefined

$\frac{0}{0}$

Step 3.1.3.4

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

Undefined

$\frac{0}{0}$

Step 3.1.4

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

Undefined

$\frac{0}{0}$

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.

$lim_{x \to 0} \frac{- 2 \cos (x) \sin (x) - 6 \sin (x)}{- \sin (x)} = lim_{x \to 0} \frac{\frac{d}{d x} [- 2 \cos (x) \sin (x) - 6 \sin (x)]}{\frac{d}{d x} [- \sin (x)]}$

Step 3.3

Find the derivative of the numerator and denominator.

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Step 3.3.1

Differentiate the numerator and denominator.

$lim_{x \to 0} \frac{\frac{d}{d x} [- 2 \cos (x) \sin (x) - 6 \sin (x)]}{\frac{d}{d x} [- \sin (x)]}$

Step 3.3.2

By the Sum Rule, the derivative of $- 2 \cos (x) \sin (x) - 6 \sin (x)$ with respect to $x$ is $\frac{d}{d x} [- 2 \cos (x) \sin (x)] + \frac{d}{d x} [- 6 \sin (x)]$ .

$lim_{x \to 0} \frac{\frac{d}{d x} [- 2 \cos (x) \sin (x)] + \frac{d}{d x} [- 6 \sin (x)]}{\frac{d}{d x} [- \sin (x)]}$

Step 3.3.3

Evaluate $\frac{d}{d x} [- 2 \cos (x) \sin (x)]$ .

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Step 3.3.3.1

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

$lim_{x \to 0} \frac{- 2 \frac{d}{d x} [\cos (x) \sin (x)] + \frac{d}{d x} [- 6 \sin (x)]}{\frac{d}{d x} [- \sin (x)]}$

Step 3.3.3.2

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 $g (x) = \sin (x)$ .

$lim_{x \to 0} \frac{- 2 (\cos (x) \frac{d}{d x} [\sin (x)] + \sin (x) \frac{d}{d x} [\cos (x)]) + \frac{d}{d x} [- 6 \sin (x)]}{\frac{d}{d x} [- \sin (x)]}$

Step 3.3.3.3

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

$lim_{x \to 0} \frac{- 2 (\cos (x) \cos (x) + \sin (x) \frac{d}{d x} [\cos (x)]) + \frac{d}{d x} [- 6 \sin (x)]}{\frac{d}{d x} [- \sin (x)]}$

Step 3.3.3.4

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

$lim_{x \to 0} \frac{- 2 (\cos (x) \cos (x) + \sin (x) (- \sin (x))) + \frac{d}{d x} [- 6 \sin (x)]}{\frac{d}{d x} [- \sin (x)]}$

Step 3.3.3.5

Raise $\cos (x)$ to the power of $1$ .

$lim_{x \to 0} \frac{- 2 (\cos^{1} (x) \cos (x) + \sin (x) (- \sin (x))) + \frac{d}{d x} [- 6 \sin (x)]}{\frac{d}{d x} [- \sin (x)]}$

Step 3.3.3.6

Raise $\cos (x)$ to the power of $1$ .

$lim_{x \to 0} \frac{- 2 (\cos^{1} (x) \cos^{1} (x) + \sin (x) (- \sin (x))) + \frac{d}{d x} [- 6 \sin (x)]}{\frac{d}{d x} [- \sin (x)]}$

Step 3.3.3.7

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

$lim_{x \to 0} \frac{- 2 ({\cos (x)}^{1 + 1} + \sin (x) (- \sin (x))) + \frac{d}{d x} [- 6 \sin (x)]}{\frac{d}{d x} [- \sin (x)]}$

Step 3.3.3.8

Add $1$ and $1$ .

$lim_{x \to 0} \frac{- 2 (\cos^{2} (x) + \sin (x) (- \sin (x))) + \frac{d}{d x} [- 6 \sin (x)]}{\frac{d}{d x} [- \sin (x)]}$

Step 3.3.3.9

Raise $\sin (x)$ to the power of $1$ .

$lim_{x \to 0} \frac{- 2 (\cos^{2} (x) - (\sin^{1} (x) \sin (x))) + \frac{d}{d x} [- 6 \sin (x)]}{\frac{d}{d x} [- \sin (x)]}$

Step 3.3.3.10

Raise $\sin (x)$ to the power of $1$ .

$lim_{x \to 0} \frac{- 2 (\cos^{2} (x) - (\sin^{1} (x) \sin^{1} (x))) + \frac{d}{d x} [- 6 \sin (x)]}{\frac{d}{d x} [- \sin (x)]}$

Step 3.3.3.11

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

$lim_{x \to 0} \frac{- 2 (\cos^{2} (x) - {\sin (x)}^{1 + 1}) + \frac{d}{d x} [- 6 \sin (x)]}{\frac{d}{d x} [- \sin (x)]}$

Step 3.3.3.12

Add $1$ and $1$ .

$lim_{x \to 0} \frac{- 2 (\cos^{2} (x) - \sin^{2} (x)) + \frac{d}{d x} [- 6 \sin (x)]}{\frac{d}{d x} [- \sin (x)]}$

Step 3.3.4

Evaluate $\frac{d}{d x} [- 6 \sin (x)]$ .

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Step 3.3.4.1

Since $- 6$ is constant with respect to $x$ , the derivative of $- 6 \sin (x)$ with respect to $x$ is $- 6 \frac{d}{d x} [\sin (x)]$ .

$lim_{x \to 0} \frac{- 2 (\cos^{2} (x) - \sin^{2} (x)) - 6 \frac{d}{d x} [\sin (x)]}{\frac{d}{d x} [- \sin (x)]}$

Step 3.3.4.2

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

$lim_{x \to 0} \frac{- 2 (\cos^{2} (x) - \sin^{2} (x)) - 6 \cos (x)}{\frac{d}{d x} [- \sin (x)]}$

Step 3.3.5

Simplify.

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Step 3.3.5.1

Apply the distributive property.

$lim_{x \to 0} \frac{- 2 \cos^{2} (x) - 2 (- \sin^{2} (x)) - 6 \cos (x)}{\frac{d}{d x} [- \sin (x)]}$

Step 3.3.5.2

Multiply $- 1$ by $- 2$ .

$lim_{x \to 0} \frac{- 2 \cos^{2} (x) + 2 \sin^{2} (x) - 6 \cos (x)}{\frac{d}{d x} [- \sin (x)]}$

Step 3.3.6

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

$lim_{x \to 0} \frac{- 2 \cos^{2} (x) + 2 \sin^{2} (x) - 6 \cos (x)}{- \frac{d}{d x} [\sin (x)]}$

Step 3.3.7

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

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

Step 4

Evaluate the limit.

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Step 4.1

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

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

Step 4.2

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

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

Step 4.3

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

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

Step 4.4

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

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

Step 4.5

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

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

Step 4.6

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

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

Step 4.7

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

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

Step 4.8

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

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

Step 4.9

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

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

Step 4.10

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

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

Step 4.11

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

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

Step 4.12

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

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

Step 5

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

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Step 5.1

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

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

Step 5.2

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

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

Step 5.3

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

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

Step 5.4

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

$\frac{- 2 \cos^{2} (0) + 2 \sin^{2} (0) - 6 \cos (0)}{- \cos (0)}$

Step 6

Simplify the answer.

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Step 6.1

Simplify the numerator.

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Step 6.1.1

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

$\frac{- 2 \cdot 1^{2} + 2 \sin^{2} (0) - 6 \cos (0)}{- \cos (0)}$

Step 6.1.2

One to any power is one.

$\frac{- 2 \cdot 1 + 2 \sin^{2} (0) - 6 \cos (0)}{- \cos (0)}$

Step 6.1.3

Multiply $- 2$ by $1$ .

$\frac{- 2 + 2 \sin^{2} (0) - 6 \cos (0)}{- \cos (0)}$

Step 6.1.4

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

$\frac{- 2 + 2 \cdot 0^{2} - 6 \cos (0)}{- \cos (0)}$

Step 6.1.5

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

$\frac{- 2 + 2 \cdot 0 - 6 \cos (0)}{- \cos (0)}$

Step 6.1.6

Multiply $2$ by $0$ .

$\frac{- 2 + 0 - 6 \cos (0)}{- \cos (0)}$

Step 6.1.7

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

$\frac{- 2 + 0 - 6 \cdot 1}{- \cos (0)}$

Step 6.1.8

Multiply $- 6$ by $1$ .

$\frac{- 2 + 0 - 6}{- \cos (0)}$

Step 6.1.9

Add $- 2$ and $0$ .

$\frac{- 2 - 6}{- \cos (0)}$

Step 6.1.10

Subtract $6$ from $- 2$ .

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

Step 6.2

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

$\frac{- 8}{- 1 \cdot 1}$

Step 6.3

Multiply $- 1$ by $1$ .

$\frac{- 8}{- 1}$

Step 6.4

Move the negative one from the denominator of $\frac{- 8}{- 1}$ .

$- 1 \cdot - 8$

Step 6.5

Multiply $- 1$ by $- 8$ .

$8$