Calculus Examples

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

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.

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

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 $2$ .

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

Step 1.2.2

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

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

Step 1.2.3

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

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

Step 1.2.4

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

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

Step 1.2.5

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

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

Step 1.2.6

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

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

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 $x$ .

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

Step 1.2.7.2

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

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

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

Multiply $- 1 \cdot 2 \cdot 2$ .

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Step 1.2.8.1.1.1

Multiply $- 1$ by $2$ .

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

Step 1.2.8.1.1.2

Multiply $- 2$ by $2$ .

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

Step 1.2.8.1.2

Subtract $4$ from $4$ .

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

Step 1.2.8.1.3

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

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

Step 1.2.8.1.4

Multiply $2$ by $1$ .

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

Step 1.2.8.2

Subtract $2$ from $2$ .

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

Step 1.3

Evaluate the limit of the denominator.

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Step 1.3.1

Evaluate the limit.

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Step 1.3.1.1

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

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

Step 1.3.1.2

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

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

Step 1.3.1.3

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

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

Step 1.3.2

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

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

Step 1.3.3

Simplify the answer.

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Step 1.3.3.1

Simplify each term.

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Step 1.3.3.1.1

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

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

Step 1.3.3.1.2

Multiply $- 1$ by $4$ .

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

Step 1.3.3.2

Subtract $4$ from $4$ .

$\frac{0}{0}$

Step 1.3.3.3

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

Undefined

$\frac{0}{0}$

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.

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

Step 3

Find the derivative of the numerator and denominator.

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Step 3.1

Differentiate the numerator and denominator.

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

Step 3.2

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

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

Step 3.3

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

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Step 3.3.1

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

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

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 $g (x) = 4 - 2 x$ .

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Step 3.3.2.1

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

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

Step 3.3.2.2

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

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

Step 3.3.2.3

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

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

Step 3.3.3

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

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

Step 3.3.4

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

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

Step 3.3.5

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

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

Step 3.3.6

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

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

Step 3.3.7

Multiply $- 2$ by $1$ .

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

Step 3.3.8

Subtract $2$ from $0$ .

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

Step 3.3.9

Multiply $- 2$ by $- 1$ .

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

Step 3.3.10

Multiply $2$ by $2$ .

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

Step 3.4

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

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Step 3.4.1

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

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

Step 3.4.2

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

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

Step 3.4.3

Multiply $- 1$ by $1$ .

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

Step 3.5

Reorder terms.

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

Step 3.6

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

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

Step 3.7

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

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

Step 3.8

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

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

Step 3.9

Add $2 x$ and $0$ .

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

Step 4

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

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

Step 5

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

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

Step 6

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

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

Step 7

Evaluate the limit of $1$ which is constant as $x$ approaches $2$ .

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

Step 8

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

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

Step 9

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

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

Step 10

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

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

Step 11

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

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

Step 12

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

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

Step 13

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

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Step 13.1

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

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

Step 13.2

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

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

Step 14

Simplify the answer.

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Step 14.1

Simplify the numerator.

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Step 14.1.1

Multiply $- 1 \cdot 2 \cdot 2$ .

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Step 14.1.1.1

Multiply $- 1$ by $2$ .

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

Step 14.1.1.2

Multiply $- 2$ by $2$ .

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

Step 14.1.2

Subtract $4$ from $4$ .

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

Step 14.1.3

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

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

Step 14.1.4

Multiply $4$ by $0$ .

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

Step 14.1.5

Add $- 1$ and $0$ .

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

Step 14.2

Move the negative in front of the fraction.

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

Step 14.3

Multiply $\frac{1}{2} (- \frac{1}{2})$ .

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Step 14.3.1

Multiply $\frac{1}{2}$ by $\frac{1}{2}$ .

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

Step 14.3.2

Multiply $2$ by $2$ .

$- \frac{1}{4}$