Calculus Examples

$lim_{x \to - 2} \frac{2}{x + 2} - \frac{x - 6}{x^{2} - 4}$

Step 1

Combine terms.

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Step 1.1

To write $\frac{2}{x + 2}$ as a fraction with a common denominator, multiply by $\frac{x^{2} - 4}{x^{2} - 4}$ .

$lim_{x \to - 2} \frac{2}{x + 2} \cdot \frac{x^{2} - 4}{x^{2} - 4} - \frac{x - 6}{x^{2} - 4}$

Step 1.2

To write $- \frac{x - 6}{x^{2} - 4}$ as a fraction with a common denominator, multiply by $\frac{x + 2}{x + 2}$ .

$lim_{x \to - 2} \frac{2}{x + 2} \cdot \frac{x^{2} - 4}{x^{2} - 4} - \frac{x - 6}{x^{2} - 4} \cdot \frac{x + 2}{x + 2}$

Step 1.3

Write each expression with a common denominator of $(x + 2) (x^{2} - 4)$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.3.1

Multiply $\frac{2}{x + 2}$ by $\frac{x^{2} - 4}{x^{2} - 4}$ .

$lim_{x \to - 2} \frac{2 (x^{2} - 4)}{(x + 2) (x^{2} - 4)} - \frac{x - 6}{x^{2} - 4} \cdot \frac{x + 2}{x + 2}$

Step 1.3.2

Multiply $\frac{x - 6}{x^{2} - 4}$ by $\frac{x + 2}{x + 2}$ .

$lim_{x \to - 2} \frac{2 (x^{2} - 4)}{(x + 2) (x^{2} - 4)} - \frac{(x - 6) (x + 2)}{(x^{2} - 4) (x + 2)}$

Step 1.3.3

Reorder the factors of $(x^{2} - 4) (x + 2)$ .

$lim_{x \to - 2} \frac{2 (x^{2} - 4)}{(x + 2) (x^{2} - 4)} - \frac{(x - 6) (x + 2)}{(x + 2) (x^{2} - 4)}$

Step 1.4

Combine the numerators over the common denominator.

$lim_{x \to - 2} \frac{2 (x^{2} - 4) - (x - 6) (x + 2)}{(x + 2) (x^{2} - 4)}$

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.

$\frac{lim_{x \to - 2} 2 (x^{2} - 4) - (x - 6) (x + 2)}{lim_{x \to - 2} (x + 2) (x^{2} - 4)}$

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 $- 2$ .

$\frac{lim_{x \to - 2} 2 (x^{2} - 4) - lim_{x \to - 2} (x - 6) (x + 2)}{lim_{x \to - 2} (x + 2) (x^{2} - 4)}$

Step 2.1.2.2

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

$\frac{2 lim_{x \to - 2} x^{2} - 4 - lim_{x \to - 2} (x - 6) (x + 2)}{lim_{x \to - 2} (x + 2) (x^{2} - 4)}$

Step 2.1.2.3

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

$\frac{2 (lim_{x \to - 2} x^{2} - lim_{x \to - 2} 4) - lim_{x \to - 2} (x - 6) (x + 2)}{lim_{x \to - 2} (x + 2) (x^{2} - 4)}$

Step 2.1.2.4

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

$\frac{2 ({(lim_{x \to - 2} x)}^{2} - lim_{x \to - 2} 4) - lim_{x \to - 2} (x - 6) (x + 2)}{lim_{x \to - 2} (x + 2) (x^{2} - 4)}$

Step 2.1.2.5

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

$\frac{2 ({(lim_{x \to - 2} x)}^{2} - 1 \cdot 4) - lim_{x \to - 2} (x - 6) (x + 2)}{lim_{x \to - 2} (x + 2) (x^{2} - 4)}$

Step 2.1.2.6

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

$\frac{2 ({(lim_{x \to - 2} x)}^{2} - 1 \cdot 4) - lim_{x \to - 2} x - 6 \cdot lim_{x \to - 2} x + 2}{lim_{x \to - 2} (x + 2) (x^{2} - 4)}$

Step 2.1.2.7

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

$\frac{2 ({(lim_{x \to - 2} x)}^{2} - 1 \cdot 4) - (lim_{x \to - 2} x - lim_{x \to - 2} 6) \cdot lim_{x \to - 2} x + 2}{lim_{x \to - 2} (x + 2) (x^{2} - 4)}$

Step 2.1.2.8

Evaluate the limit of $6$ which is constant as $x$ approaches $- 2$ .

$\frac{2 ({(lim_{x \to - 2} x)}^{2} - 1 \cdot 4) - (lim_{x \to - 2} x - 1 \cdot 6) \cdot lim_{x \to - 2} x + 2}{lim_{x \to - 2} (x + 2) (x^{2} - 4)}$

Step 2.1.2.9

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

$\frac{2 ({(lim_{x \to - 2} x)}^{2} - 1 \cdot 4) - (lim_{x \to - 2} x - 1 \cdot 6) \cdot (lim_{x \to - 2} x + lim_{x \to - 2} 2)}{lim_{x \to - 2} (x + 2) (x^{2} - 4)}$

Step 2.1.2.10

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

$\frac{2 ({(lim_{x \to - 2} x)}^{2} - 1 \cdot 4) - (lim_{x \to - 2} x - 1 \cdot 6) \cdot (lim_{x \to - 2} x + 2)}{lim_{x \to - 2} (x + 2) (x^{2} - 4)}$

Step 2.1.2.11

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

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Step 2.1.2.11.1

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

$\frac{2 ({(- 2)}^{2} - 1 \cdot 4) - (lim_{x \to - 2} x - 1 \cdot 6) \cdot (lim_{x \to - 2} x + 2)}{lim_{x \to - 2} (x + 2) (x^{2} - 4)}$

Step 2.1.2.11.2

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

$\frac{2 ({(- 2)}^{2} - 1 \cdot 4) - (- 2 - 1 \cdot 6) \cdot (lim_{x \to - 2} x + 2)}{lim_{x \to - 2} (x + 2) (x^{2} - 4)}$

Step 2.1.2.11.3

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

$\frac{2 ({(- 2)}^{2} - 1 \cdot 4) - (- 2 - 1 \cdot 6) \cdot (- 2 + 2)}{lim_{x \to - 2} (x + 2) (x^{2} - 4)}$

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

Simplify each term.

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Step 2.1.2.12.1.1.1

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

$\frac{2 (4 - 1 \cdot 4) - (- 2 - 1 \cdot 6) \cdot (- 2 + 2)}{lim_{x \to - 2} (x + 2) (x^{2} - 4)}$

Step 2.1.2.12.1.1.2

Multiply $- 1$ by $4$ .

$\frac{2 (4 - 4) - (- 2 - 1 \cdot 6) \cdot (- 2 + 2)}{lim_{x \to - 2} (x + 2) (x^{2} - 4)}$

Step 2.1.2.12.1.2

Subtract $4$ from $4$ .

$\frac{2 \cdot 0 - (- 2 - 1 \cdot 6) \cdot (- 2 + 2)}{lim_{x \to - 2} (x + 2) (x^{2} - 4)}$

Step 2.1.2.12.1.3

Multiply $2$ by $0$ .

$\frac{0 - (- 2 - 1 \cdot 6) \cdot (- 2 + 2)}{lim_{x \to - 2} (x + 2) (x^{2} - 4)}$

Step 2.1.2.12.1.4

Multiply $- 1$ by $6$ .

$\frac{0 - (- 2 - 6) \cdot (- 2 + 2)}{lim_{x \to - 2} (x + 2) (x^{2} - 4)}$

Step 2.1.2.12.1.5

Subtract $6$ from $- 2$ .

$\frac{0 - - 8 \cdot (- 2 + 2)}{lim_{x \to - 2} (x + 2) (x^{2} - 4)}$

Step 2.1.2.12.1.6

Multiply $- 1$ by $- 8$ .

$\frac{0 + 8 \cdot (- 2 + 2)}{lim_{x \to - 2} (x + 2) (x^{2} - 4)}$

Step 2.1.2.12.1.7

Add $- 2$ and $2$ .

$\frac{0 + 8 \cdot 0}{lim_{x \to - 2} (x + 2) (x^{2} - 4)}$

Step 2.1.2.12.1.8

Multiply $8$ by $0$ .

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

Step 2.1.2.12.2

Add $0$ and $0$ .

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

Step 2.1.3

Evaluate the limit of the denominator.

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Step 2.1.3.1

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

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

Step 2.1.3.2

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

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

Step 2.1.3.3

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

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

Step 2.1.3.4

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

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

Step 2.1.3.5

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

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

Step 2.1.3.6

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

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

Step 2.1.3.7

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

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Step 2.1.3.7.1

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

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

Step 2.1.3.7.2

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

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

Step 2.1.3.8

Simplify the answer.

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Step 2.1.3.8.1

Add $- 2$ and $2$ .

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

Step 2.1.3.8.2

Simplify each term.

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Step 2.1.3.8.2.1

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

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

Step 2.1.3.8.2.2

Multiply $- 1$ by $4$ .

$\frac{0}{0 \cdot (4 - 4)}$

Step 2.1.3.8.3

Subtract $4$ from $4$ .

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

Step 2.1.3.8.4

Multiply $0$ by $0$ .

$\frac{0}{0}$

Step 2.1.3.8.5

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

Undefined

$\frac{0}{0}$

Step 2.1.3.9

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.

$lim_{x \to - 2} \frac{2 (x^{2} - 4) - (x - 6) (x + 2)}{(x + 2) (x^{2} - 4)} = lim_{x \to - 2} \frac{\frac{d}{d x} [2 (x^{2} - 4) - (x - 6) (x + 2)]}{\frac{d}{d x} [(x + 2) (x^{2} - 4)]}$

Step 2.3

Find the derivative of the numerator and denominator.

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Step 2.3.1

Differentiate the numerator and denominator.

$lim_{x \to - 2} \frac{\frac{d}{d x} [2 (x^{2} - 4) - (x - 6) (x + 2)]}{\frac{d}{d x} [(x + 2) (x^{2} - 4)]}$

Step 2.3.2

By the Sum Rule, the derivative of $2 (x^{2} - 4) - (x - 6) (x + 2)$ with respect to $x$ is $\frac{d}{d x} [2 (x^{2} - 4)] + \frac{d}{d x} [- (x - 6) (x + 2)]$ .

$lim_{x \to - 2} \frac{\frac{d}{d x} [2 (x^{2} - 4)] + \frac{d}{d x} [- (x - 6) (x + 2)]}{\frac{d}{d x} [(x + 2) (x^{2} - 4)]}$

Step 2.3.3

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

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Step 2.3.3.1

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

$lim_{x \to - 2} \frac{2 \frac{d}{d x} [x^{2} - 4] + \frac{d}{d x} [- (x - 6) (x + 2)]}{\frac{d}{d x} [(x + 2) (x^{2} - 4)]}$

Step 2.3.3.2

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

$lim_{x \to - 2} \frac{2 (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 4]) + \frac{d}{d x} [- (x - 6) (x + 2)]}{\frac{d}{d x} [(x + 2) (x^{2} - 4)]}$

Step 2.3.3.3

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

$lim_{x \to - 2} \frac{2 (2 x + \frac{d}{d x} [- 4]) + \frac{d}{d x} [- (x - 6) (x + 2)]}{\frac{d}{d x} [(x + 2) (x^{2} - 4)]}$

Step 2.3.3.4

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

$lim_{x \to - 2} \frac{2 (2 x + 0) + \frac{d}{d x} [- (x - 6) (x + 2)]}{\frac{d}{d x} [(x + 2) (x^{2} - 4)]}$

Step 2.3.3.5

Add $2 x$ and $0$ .

$lim_{x \to - 2} \frac{2 (2 x) + \frac{d}{d x} [- (x - 6) (x + 2)]}{\frac{d}{d x} [(x + 2) (x^{2} - 4)]}$

Step 2.3.3.6

Multiply $2$ by $2$ .

$lim_{x \to - 2} \frac{4 x + \frac{d}{d x} [- (x - 6) (x + 2)]}{\frac{d}{d x} [(x + 2) (x^{2} - 4)]}$

Step 2.3.4

Evaluate $\frac{d}{d x} [- (x - 6) (x + 2)]$ .

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Step 2.3.4.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- (x - 6) (x + 2)$ with respect to $x$ is $- \frac{d}{d x} [(x - 6) (x + 2)]$ .

$lim_{x \to - 2} \frac{4 x - \frac{d}{d x} [(x - 6) (x + 2)]}{\frac{d}{d x} [(x + 2) (x^{2} - 4)]}$

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 $f (x) = x - 6$ and $g (x) = x + 2$ .

$lim_{x \to - 2} \frac{4 x - ((x - 6) \frac{d}{d x} [x + 2] + (x + 2) \frac{d}{d x} [x - 6])}{\frac{d}{d x} [(x + 2) (x^{2} - 4)]}$

Step 2.3.4.3

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

$lim_{x \to - 2} \frac{4 x - ((x - 6) (\frac{d}{d x} [x] + \frac{d}{d x} [2]) + (x + 2) \frac{d}{d x} [x - 6])}{\frac{d}{d x} [(x + 2) (x^{2} - 4)]}$

Step 2.3.4.4

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

$lim_{x \to - 2} \frac{4 x - ((x - 6) (1 + \frac{d}{d x} [2]) + (x + 2) \frac{d}{d x} [x - 6])}{\frac{d}{d x} [(x + 2) (x^{2} - 4)]}$

Step 2.3.4.5

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

$lim_{x \to - 2} \frac{4 x - ((x - 6) (1 + 0) + (x + 2) \frac{d}{d x} [x - 6])}{\frac{d}{d x} [(x + 2) (x^{2} - 4)]}$

Step 2.3.4.6

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

$lim_{x \to - 2} \frac{4 x - ((x - 6) (1 + 0) + (x + 2) (\frac{d}{d x} [x] + \frac{d}{d x} [- 6]))}{\frac{d}{d x} [(x + 2) (x^{2} - 4)]}$

Step 2.3.4.7

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

$lim_{x \to - 2} \frac{4 x - ((x - 6) (1 + 0) + (x + 2) (1 + \frac{d}{d x} [- 6]))}{\frac{d}{d x} [(x + 2) (x^{2} - 4)]}$

Step 2.3.4.8

Since $- 6$ is constant with respect to $x$ , the derivative of $- 6$ with respect to $x$ is $0$ .

$lim_{x \to - 2} \frac{4 x - ((x - 6) (1 + 0) + (x + 2) (1 + 0))}{\frac{d}{d x} [(x + 2) (x^{2} - 4)]}$

Step 2.3.4.9

Add $1$ and $0$ .

$lim_{x \to - 2} \frac{4 x - ((x - 6) \cdot 1 + (x + 2) (1 + 0))}{\frac{d}{d x} [(x + 2) (x^{2} - 4)]}$

Step 2.3.4.10

Multiply $x - 6$ by $1$ .

$lim_{x \to - 2} \frac{4 x - (x - 6 + (x + 2) (1 + 0))}{\frac{d}{d x} [(x + 2) (x^{2} - 4)]}$

Step 2.3.4.11

Add $1$ and $0$ .

$lim_{x \to - 2} \frac{4 x - (x - 6 + (x + 2) \cdot 1)}{\frac{d}{d x} [(x + 2) (x^{2} - 4)]}$

Step 2.3.4.12

Multiply $x + 2$ by $1$ .

$lim_{x \to - 2} \frac{4 x - (x - 6 + x + 2)}{\frac{d}{d x} [(x + 2) (x^{2} - 4)]}$

Step 2.3.4.13

Add $x$ and $x$ .

$lim_{x \to - 2} \frac{4 x - (2 x - 6 + 2)}{\frac{d}{d x} [(x + 2) (x^{2} - 4)]}$

Step 2.3.4.14

Add $- 6$ and $2$ .

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

Step 2.3.5

Simplify.

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Step 2.3.5.1

Apply the distributive property.

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

Step 2.3.5.2

Combine terms.

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Step 2.3.5.2.1

Multiply $2$ by $- 1$ .

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

Step 2.3.5.2.2

Multiply $- 1$ by $- 4$ .

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

Step 2.3.5.2.3

Subtract $2 x$ from $4 x$ .

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

Step 2.3.6

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 $f (x) = x + 2$ and $g (x) = x^{2} - 4$ .

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

Step 2.3.7

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

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

Step 2.3.8

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

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

Step 2.3.9

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

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

Step 2.3.10

Add $2 x$ and $0$ .

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

Step 2.3.11

Move $2$ to the left of $x + 2$ .

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

Step 2.3.12

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

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

Step 2.3.13

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

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

Step 2.3.14

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

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

Step 2.3.15

Add $1$ and $0$ .

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

Step 2.3.16

Multiply $x^{2} - 4$ by $1$ .

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

Step 2.3.17

Simplify.

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Step 2.3.17.1

Apply the distributive property.

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

Step 2.3.17.2

Apply the distributive property.

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

Step 2.3.17.3

Combine terms.

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Step 2.3.17.3.1

Raise $x$ to the power of $1$ .

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

Step 2.3.17.3.2

Raise $x$ to the power of $1$ .

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

Step 2.3.17.3.3

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

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

Step 2.3.17.3.4

Add $1$ and $1$ .

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

Step 2.3.17.3.5

Multiply $2$ by $2$ .

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

Step 2.3.17.3.6

Add $2 x^{2}$ and $x^{2}$ .

$lim_{x \to - 2} \frac{2 x + 4}{3 x^{2} + 4 x - 4}$

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.

$\frac{lim_{x \to - 2} 2 x + 4}{lim_{x \to - 2} 3 x^{2} + 4 x - 4}$

Step 3.1.2

Evaluate the limit of the numerator.

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Step 3.1.2.1

Evaluate the limit.

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Step 3.1.2.1.1

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

$\frac{lim_{x \to - 2} 2 x + lim_{x \to - 2} 4}{lim_{x \to - 2} 3 x^{2} + 4 x - 4}$

Step 3.1.2.1.2

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

$\frac{2 lim_{x \to - 2} x + lim_{x \to - 2} 4}{lim_{x \to - 2} 3 x^{2} + 4 x - 4}$

Step 3.1.2.1.3

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

$\frac{2 lim_{x \to - 2} x + 4}{lim_{x \to - 2} 3 x^{2} + 4 x - 4}$

Step 3.1.2.2

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

$\frac{2 \cdot - 2 + 4}{lim_{x \to - 2} 3 x^{2} + 4 x - 4}$

Step 3.1.2.3

Simplify the answer.

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Step 3.1.2.3.1

Multiply $2$ by $- 2$ .

$\frac{- 4 + 4}{lim_{x \to - 2} 3 x^{2} + 4 x - 4}$

Step 3.1.2.3.2

Add $- 4$ and $4$ .

$\frac{0}{lim_{x \to - 2} 3 x^{2} + 4 x - 4}$

Step 3.1.3

Evaluate the limit of the denominator.

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Step 3.1.3.1

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

$\frac{0}{lim_{x \to - 2} 3 x^{2} + lim_{x \to - 2} 4 x - lim_{x \to - 2} 4}$

Step 3.1.3.2

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

$\frac{0}{3 lim_{x \to - 2} x^{2} + lim_{x \to - 2} 4 x - lim_{x \to - 2} 4}$

Step 3.1.3.3

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

$\frac{0}{3 {(lim_{x \to - 2} x)}^{2} + lim_{x \to - 2} 4 x - lim_{x \to - 2} 4}$

Step 3.1.3.4

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

$\frac{0}{3 {(lim_{x \to - 2} x)}^{2} + 4 lim_{x \to - 2} x - lim_{x \to - 2} 4}$

Step 3.1.3.5

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

$\frac{0}{3 {(lim_{x \to - 2} x)}^{2} + 4 lim_{x \to - 2} x - 1 \cdot 4}$

Step 3.1.3.6

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

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Step 3.1.3.6.1

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

$\frac{0}{3 {(- 2)}^{2} + 4 lim_{x \to - 2} x - 1 \cdot 4}$

Step 3.1.3.6.2

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

$\frac{0}{3 {(- 2)}^{2} + 4 \cdot - 2 - 1 \cdot 4}$

Step 3.1.3.7

Simplify the answer.

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Step 3.1.3.7.1

Simplify each term.

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Step 3.1.3.7.1.1

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

$\frac{0}{3 \cdot 4 + 4 \cdot - 2 - 1 \cdot 4}$

Step 3.1.3.7.1.2

Multiply $3$ by $4$ .

$\frac{0}{12 + 4 \cdot - 2 - 1 \cdot 4}$

Step 3.1.3.7.1.3

Multiply $4$ by $- 2$ .

$\frac{0}{12 - 8 - 1 \cdot 4}$

Step 3.1.3.7.1.4

Multiply $- 1$ by $4$ .

$\frac{0}{12 - 8 - 4}$

Step 3.1.3.7.2

Subtract $8$ from $12$ .

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

Step 3.1.3.7.3

Subtract $4$ from $4$ .

$\frac{0}{0}$

Step 3.1.3.7.4

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

Undefined

$\frac{0}{0}$

Step 3.1.3.8

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.

$lim_{x \to - 2} \frac{2 x + 4}{3 x^{2} + 4 x - 4} = lim_{x \to - 2} \frac{\frac{d}{d x} [2 x + 4]}{\frac{d}{d x} [3 x^{2} + 4 x - 4]}$

Step 3.3

Find the derivative of the numerator and denominator.

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Step 3.3.1

Differentiate the numerator and denominator.

$lim_{x \to - 2} \frac{\frac{d}{d x} [2 x + 4]}{\frac{d}{d x} [3 x^{2} + 4 x - 4]}$

Step 3.3.2

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

$lim_{x \to - 2} \frac{\frac{d}{d x} [2 x] + \frac{d}{d x} [4]}{\frac{d}{d x} [3 x^{2} + 4 x - 4]}$

Step 3.3.3

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

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Step 3.3.3.1

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

$lim_{x \to - 2} \frac{2 \frac{d}{d x} [x] + \frac{d}{d x} [4]}{\frac{d}{d x} [3 x^{2} + 4 x - 4]}$

Step 3.3.3.2

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

$lim_{x \to - 2} \frac{2 \cdot 1 + \frac{d}{d x} [4]}{\frac{d}{d x} [3 x^{2} + 4 x - 4]}$

Step 3.3.3.3

Multiply $2$ by $1$ .

$lim_{x \to - 2} \frac{2 + \frac{d}{d x} [4]}{\frac{d}{d x} [3 x^{2} + 4 x - 4]}$

Step 3.3.4

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

$lim_{x \to - 2} \frac{2 + 0}{\frac{d}{d x} [3 x^{2} + 4 x - 4]}$

Step 3.3.5

Add $2$ and $0$ .

$lim_{x \to - 2} \frac{2}{\frac{d}{d x} [3 x^{2} + 4 x - 4]}$

Step 3.3.6

By the Sum Rule, the derivative of $3 x^{2} + 4 x - 4$ with respect to $x$ is $\frac{d}{d x} [3 x^{2}] + \frac{d}{d x} [4 x] + \frac{d}{d x} [- 4]$ .

$lim_{x \to - 2} \frac{2}{\frac{d}{d x} [3 x^{2}] + \frac{d}{d x} [4 x] + \frac{d}{d x} [- 4]}$

Step 3.3.7

Evaluate $\frac{d}{d x} [3 x^{2}]$ .

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Step 3.3.7.1

Since $3$ is constant with respect to $x$ , the derivative of $3 x^{2}$ with respect to $x$ is $3 \frac{d}{d x} [x^{2}]$ .

$lim_{x \to - 2} \frac{2}{3 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [4 x] + \frac{d}{d x} [- 4]}$

Step 3.3.7.2

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

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

Step 3.3.7.3

Multiply $2$ by $3$ .

$lim_{x \to - 2} \frac{2}{6 x + \frac{d}{d x} [4 x] + \frac{d}{d x} [- 4]}$

Step 3.3.8

Evaluate $\frac{d}{d x} [4 x]$ .

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Step 3.3.8.1

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

$lim_{x \to - 2} \frac{2}{6 x + 4 \frac{d}{d x} [x] + \frac{d}{d x} [- 4]}$

Step 3.3.8.2

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

$lim_{x \to - 2} \frac{2}{6 x + 4 \cdot 1 + \frac{d}{d x} [- 4]}$

Step 3.3.8.3

Multiply $4$ by $1$ .

$lim_{x \to - 2} \frac{2}{6 x + 4 + \frac{d}{d x} [- 4]}$

Step 3.3.9

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

$lim_{x \to - 2} \frac{2}{6 x + 4 + 0}$

Step 3.3.10

Add $6 x + 4$ and $0$ .

$lim_{x \to - 2} \frac{2}{6 x + 4}$

Step 3.4

Cancel the common factor of $2$ and $6 x + 4$ .

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Step 3.4.1

Factor $2$ out of $2$ .

$lim_{x \to - 2} \frac{2 (1)}{6 x + 4}$

Step 3.4.2

Cancel the common factors.

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Step 3.4.2.1

Factor $2$ out of $6 x$ .

$lim_{x \to - 2} \frac{2 (1)}{2 (3 x) + 4}$

Step 3.4.2.2

Factor $2$ out of $4$ .

$lim_{x \to - 2} \frac{2 \cdot 1}{2 (3 x) + 2 \cdot 2}$

Step 3.4.2.3

Factor $2$ out of $2 (3 x) + 2 (2)$ .

$lim_{x \to - 2} \frac{2 (1)}{2 (3 x + 2)}$

Step 3.4.2.4

Cancel the common factor.

$lim_{x \to - 2} \frac{2 \cdot 1}{2 (3 x + 2)}$

Step 3.4.2.5

Rewrite the expression.

$lim_{x \to - 2} \frac{1}{3 x + 2}$

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 $- 2$ .

$\frac{lim_{x \to - 2} 1}{lim_{x \to - 2} 3 x + 2}$

Step 4.2

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

$\frac{1}{lim_{x \to - 2} 3 x + 2}$

Step 4.3

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

$\frac{1}{lim_{x \to - 2} 3 x + lim_{x \to - 2} 2}$

Step 4.4

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

$\frac{1}{3 lim_{x \to - 2} x + lim_{x \to - 2} 2}$

Step 4.5

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

$\frac{1}{3 lim_{x \to - 2} x + 2}$

Step 5

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

$\frac{1}{3 \cdot - 2 + 2}$

Step 6

Simplify the answer.

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Step 6.1

Simplify the denominator.

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Step 6.1.1

Multiply $3$ by $- 2$ .

$\frac{1}{- 6 + 2}$

Step 6.1.2

Add $- 6$ and $2$ .

$\frac{1}{- 4}$

Step 6.2

Move the negative in front of the fraction.

$- \frac{1}{4}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$- \frac{1}{4}$

Decimal Form:

$- 0.25$