Calculus Examples

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

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.

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

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 $0$ .

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

Step 1.2.2

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

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

Step 1.2.3

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

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

Step 1.2.4

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

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

Step 1.2.5

Move the term $7$ outside of the limit because it is constant with respect to $x$ .

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

Step 1.2.6

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

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Step 1.2.6.1

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

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

Step 1.2.6.2

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

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

Step 1.2.7

Simplify the answer.

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Step 1.2.7.1

Simplify each term.

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Step 1.2.7.1.1

Multiply $6$ by $0$ .

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

Step 1.2.7.1.2

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

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

Step 1.2.7.1.3

Multiply $7$ by $0$ .

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

Step 1.2.7.1.4

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

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

Step 1.2.7.1.5

Multiply $- 1$ by $1$ .

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

Step 1.2.7.2

Subtract $1$ from $1$ .

$\frac{0}{lim_{x \to 0} x^{2}}$

Step 1.3

Evaluate the limit of the denominator.

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Step 1.3.1

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

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

Step 1.3.2

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

$\frac{0}{0^{2}}$

Step 1.3.3

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

$\frac{0}{0}$

Step 1.3.4

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.

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

Step 3

Find the derivative of the numerator and denominator.

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Step 3.1

Differentiate the numerator and denominator.

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

Step 3.2

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

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

Step 3.3

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

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Step 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 $f (x) = \cos (x)$ and $g (x) = 6 x$ .

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Step 3.3.1.1

To apply the Chain Rule, set $u_{1}$ as $6 x$ .

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

Step 3.3.1.2

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

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

Step 3.3.1.3

Replace all occurrences of $u_{1}$ with $6 x$ .

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

Step 3.3.2

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

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

Step 3.3.3

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

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

Step 3.3.4

Multiply $6$ by $1$ .

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

Step 3.3.5

Multiply $6$ by $- 1$ .

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

Step 3.4

Evaluate $\frac{d}{d x} [- \cos (7 x)]$ .

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Step 3.4.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- \cos (7 x)$ with respect to $x$ is $- \frac{d}{d x} [\cos (7 x)]$ .

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

Step 3.4.2

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 $g (x) = 7 x$ .

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Step 3.4.2.1

To apply the Chain Rule, set $u_{2}$ as $7 x$ .

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

Step 3.4.2.2

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

$lim_{x \to 0} \frac{- 6 \sin (6 x) - (- \sin (u_{2}) \frac{d}{d x} [7 x])}{\frac{d}{d x} [x^{2}]}$

Step 3.4.2.3

Replace all occurrences of $u_{2}$ with $7 x$ .

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

Step 3.4.3

Since $7$ is constant with respect to $x$ , the derivative of $7 x$ with respect to $x$ is $7 \frac{d}{d x} [x]$ .

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

Step 3.4.4

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

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

Step 3.4.5

Multiply $7$ by $1$ .

$lim_{x \to 0} \frac{- 6 \sin (6 x) - (- \sin (7 x) \cdot 7)}{\frac{d}{d x} [x^{2}]}$

Step 3.4.6

Multiply $7$ by $- 1$ .

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

Step 3.4.7

Multiply $- 7$ by $- 1$ .

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

Step 3.5

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

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

Step 4

Move the term $\frac{1}{2}$ outside of the limit because it is constant with respect to $x$ .

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

Step 5

Apply L'Hospital's rule.

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Step 5.1

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

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Step 5.1.1

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

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

Step 5.1.2

Evaluate the limit of the numerator.

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Step 5.1.2.1

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

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

Step 5.1.2.2

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

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

Step 5.1.2.3

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

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

Step 5.1.2.4

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

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

Step 5.1.2.5

Move the term $7$ outside of the limit because it is constant with respect to $x$ .

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

Step 5.1.2.6

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

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

Step 5.1.2.7

Move the term $7$ outside of the limit because it is constant with respect to $x$ .

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

Step 5.1.2.8

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

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Step 5.1.2.8.1

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

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

Step 5.1.2.8.2

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

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

Step 5.1.2.9

Simplify the answer.

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Step 5.1.2.9.1

Simplify each term.

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Step 5.1.2.9.1.1

Multiply $6$ by $0$ .

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

Step 5.1.2.9.1.2

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

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

Step 5.1.2.9.1.3

Multiply $- 6$ by $0$ .

$\frac{1}{2} \cdot \frac{0 + 7 \sin (7 \cdot 0)}{lim_{x \to 0} x}$

Step 5.1.2.9.1.4

Multiply $7$ by $0$ .

$\frac{1}{2} \cdot \frac{0 + 7 \sin (0)}{lim_{x \to 0} x}$

Step 5.1.2.9.1.5

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

$\frac{1}{2} \cdot \frac{0 + 7 \cdot 0}{lim_{x \to 0} x}$

Step 5.1.2.9.1.6

Multiply $7$ by $0$ .

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

Step 5.1.2.9.2

Add $0$ and $0$ .

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

Step 5.1.3

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

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

Step 5.1.4

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

Undefined

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

Step 5.2

Since $\frac{0}{0}$ is of indeterminate form, apply L'Hospital's Rule. L'Hospital's Rule states that the limit of a quotient of functions is equal to the limit of the quotient of their derivatives.

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

Step 5.3

Find the derivative of the numerator and denominator.

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Step 5.3.1

Differentiate the numerator and denominator.

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

Step 5.3.2

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

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

Step 5.3.3

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

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Step 5.3.3.1

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

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

Step 5.3.3.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \sin (x)$ and $g (x) = 6 x$ .

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Step 5.3.3.2.1

To apply the Chain Rule, set $u_{1}$ as $6 x$ .

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

Step 5.3.3.2.2

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

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

Step 5.3.3.2.3

Replace all occurrences of $u_{1}$ with $6 x$ .

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

Step 5.3.3.3

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

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

Step 5.3.3.4

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

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

Step 5.3.3.5

Multiply $6$ by $1$ .

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

Step 5.3.3.6

Move $6$ to the left of $\cos (6 x)$ .

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

Step 5.3.3.7

Multiply $6$ by $- 6$ .

$\frac{1}{2} lim_{x \to 0} \frac{- 36 \cos (6 x) + \frac{d}{d x} [7 \sin (7 x)]}{\frac{d}{d x} [x]}$

Step 5.3.4

Evaluate $\frac{d}{d x} [7 \sin (7 x)]$ .

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Step 5.3.4.1

Since $7$ is constant with respect to $x$ , the derivative of $7 \sin (7 x)$ with respect to $x$ is $7 \frac{d}{d x} [\sin (7 x)]$ .

$\frac{1}{2} lim_{x \to 0} \frac{- 36 \cos (6 x) + 7 \frac{d}{d x} [\sin (7 x)]}{\frac{d}{d x} [x]}$

Step 5.3.4.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \sin (x)$ and $g (x) = 7 x$ .

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Step 5.3.4.2.1

To apply the Chain Rule, set $u_{2}$ as $7 x$ .

$\frac{1}{2} lim_{x \to 0} \frac{- 36 \cos (6 x) + 7 \frac{d}{d u_{2}} [\sin (u_{2})] \frac{d}{d x} [7 x]}{\frac{d}{d x} [x]}$

Step 5.3.4.2.2

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

$\frac{1}{2} lim_{x \to 0} \frac{- 36 \cos (6 x) + 7 \cos (u_{2}) \frac{d}{d x} [7 x]}{\frac{d}{d x} [x]}$

Step 5.3.4.2.3

Replace all occurrences of $u_{2}$ with $7 x$ .

$\frac{1}{2} lim_{x \to 0} \frac{- 36 \cos (6 x) + 7 \cos (7 x) \frac{d}{d x} [7 x]}{\frac{d}{d x} [x]}$

Step 5.3.4.3

Since $7$ is constant with respect to $x$ , the derivative of $7 x$ with respect to $x$ is $7 \frac{d}{d x} [x]$ .

$\frac{1}{2} lim_{x \to 0} \frac{- 36 \cos (6 x) + 7 \cos (7 x) \cdot 7 \frac{d}{d x} [x]}{\frac{d}{d x} [x]}$

Step 5.3.4.4

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

$\frac{1}{2} lim_{x \to 0} \frac{- 36 \cos (6 x) + 7 \cos (7 x) \cdot 7 \cdot 1}{\frac{d}{d x} [x]}$

Step 5.3.4.5

Multiply $7$ by $1$ .

$\frac{1}{2} lim_{x \to 0} \frac{- 36 \cos (6 x) + 7 \cos (7 x) \cdot 7}{\frac{d}{d x} [x]}$

Step 5.3.4.6

Move $7$ to the left of $\cos (7 x)$ .

$\frac{1}{2} lim_{x \to 0} \frac{- 36 \cos (6 x) + 7 \cdot 7 \cdot \cos (7 x)}{\frac{d}{d x} [x]}$

Step 5.3.4.7

Multiply $7$ by $7$ .

$\frac{1}{2} lim_{x \to 0} \frac{- 36 \cos (6 x) + 49 \cos (7 x)}{\frac{d}{d x} [x]}$

Step 5.3.5

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

$\frac{1}{2} lim_{x \to 0} \frac{- 36 \cos (6 x) + 49 \cos (7 x)}{1}$

Step 5.4

Divide $- 36 \cos (6 x) + 49 \cos (7 x)$ by $1$ .

$\frac{1}{2} lim_{x \to 0} - 36 \cos (6 x) + 49 \cos (7 x)$

Step 6

Evaluate the limit.

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Step 6.1

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

$\frac{1}{2} (- lim_{x \to 0} 36 \cos (6 x) + lim_{x \to 0} 49 \cos (7 x))$

Step 6.2

Move the term $36$ outside of the limit because it is constant with respect to $x$ .

$\frac{1}{2} (- 36 lim_{x \to 0} \cos (6 x) + lim_{x \to 0} 49 \cos (7 x))$

Step 6.3

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

$\frac{1}{2} (- 36 \cos (lim_{x \to 0} 6 x) + lim_{x \to 0} 49 \cos (7 x))$

Step 6.4

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

$\frac{1}{2} (- 36 \cos (6 lim_{x \to 0} x) + lim_{x \to 0} 49 \cos (7 x))$

Step 6.5

Move the term $49$ outside of the limit because it is constant with respect to $x$ .

$\frac{1}{2} (- 36 \cos (6 lim_{x \to 0} x) + 49 lim_{x \to 0} \cos (7 x))$

Step 6.6

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

$\frac{1}{2} (- 36 \cos (6 lim_{x \to 0} x) + 49 \cos (lim_{x \to 0} 7 x))$

Step 6.7

Move the term $7$ outside of the limit because it is constant with respect to $x$ .

$\frac{1}{2} (- 36 \cos (6 lim_{x \to 0} x) + 49 \cos (7 lim_{x \to 0} x))$

Step 7

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

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Step 7.1

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

$\frac{1}{2} (- 36 \cos (6 \cdot 0) + 49 \cos (7 lim_{x \to 0} x))$

Step 7.2

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

$\frac{1}{2} (- 36 \cos (6 \cdot 0) + 49 \cos (7 \cdot 0))$

Step 8

Simplify the answer.

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Step 8.1

Simplify each term.

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Step 8.1.1

Multiply $6$ by $0$ .

$\frac{1}{2} (- 36 \cos (0) + 49 \cos (7 \cdot 0))$

Step 8.1.2

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

$\frac{1}{2} (- 36 \cdot 1 + 49 \cos (7 \cdot 0))$

Step 8.1.3

Multiply $- 36$ by $1$ .

$\frac{1}{2} (- 36 + 49 \cos (7 \cdot 0))$

Step 8.1.4

Multiply $7$ by $0$ .

$\frac{1}{2} (- 36 + 49 \cos (0))$

Step 8.1.5

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

$\frac{1}{2} (- 36 + 49 \cdot 1)$

Step 8.1.6

Multiply $49$ by $1$ .

$\frac{1}{2} (- 36 + 49)$

Step 8.2

Add $- 36$ and $49$ .

$\frac{1}{2} \cdot 13$

Step 8.3

Combine $\frac{1}{2}$ and $13$ .

$\frac{13}{2}$