Calculus Examples

$lim_{x \to - 3} \frac{6}{9 - x^{2}} - \frac{1}{x + 3}$

Step 1

Combine terms.

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Step 1.1

To write $\frac{6}{9 - x^{2}}$ as a fraction with a common denominator, multiply by $\frac{x + 3}{x + 3}$ .

$lim_{x \to - 3} \frac{6}{9 - x^{2}} \cdot \frac{x + 3}{x + 3} - \frac{1}{x + 3}$

Step 1.2

To write $- \frac{1}{x + 3}$ as a fraction with a common denominator, multiply by $\frac{9 - x^{2}}{9 - x^{2}}$ .

$lim_{x \to - 3} \frac{6}{9 - x^{2}} \cdot \frac{x + 3}{x + 3} - \frac{1}{x + 3} \cdot \frac{9 - x^{2}}{9 - x^{2}}$

Step 1.3

Write each expression with a common denominator of $(9 - x^{2}) (x + 3)$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.3.1

Multiply $\frac{6}{9 - x^{2}}$ by $\frac{x + 3}{x + 3}$ .

$lim_{x \to - 3} \frac{6 (x + 3)}{(9 - x^{2}) (x + 3)} - \frac{1}{x + 3} \cdot \frac{9 - x^{2}}{9 - x^{2}}$

Step 1.3.2

Multiply $\frac{1}{x + 3}$ by $\frac{9 - x^{2}}{9 - x^{2}}$ .

$lim_{x \to - 3} \frac{6 (x + 3)}{(9 - x^{2}) (x + 3)} - \frac{9 - x^{2}}{(x + 3) (9 - x^{2})}$

Step 1.3.3

Reorder the factors of $(x + 3) (9 - x^{2})$ .

$lim_{x \to - 3} \frac{6 (x + 3)}{(9 - x^{2}) (x + 3)} - \frac{9 - x^{2}}{(9 - x^{2}) (x + 3)}$

Step 1.4

Combine the numerators over the common denominator.

$lim_{x \to - 3} \frac{6 (x + 3) - (9 - x^{2})}{(9 - x^{2}) (x + 3)}$

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.

$\frac{lim_{x \to - 3} 6 (x + 3) - (9 - x^{2})}{lim_{x \to - 3} (9 - x^{2}) (x + 3)}$

Step 2.1.2

Evaluate the limits by plugging in $- 3$ for all occurrences of $x$ .

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Step 2.1.2.1

Evaluate the limit of $6 (x + 3) - (9 - x^{2})$ by plugging in $- 3$ for $x$ .

$\frac{6 (- 3 + 3) - (9 - {(- 3)}^{2})}{lim_{x \to - 3} (9 - x^{2}) (x + 3)}$

Step 2.1.2.2

Simplify each term.

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Step 2.1.2.2.1

Add $- 3$ and $3$ .

$\frac{6 \cdot 0 - (9 - {(- 3)}^{2})}{lim_{x \to - 3} (9 - x^{2}) (x + 3)}$

Step 2.1.2.2.2

Multiply $6$ by $0$ .

$\frac{0 - (9 - {(- 3)}^{2})}{lim_{x \to - 3} (9 - x^{2}) (x + 3)}$

Step 2.1.2.2.3

Simplify each term.

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Step 2.1.2.2.3.1

Raise $- 3$ to the power of $2$ .

$\frac{0 - (9 - 1 \cdot 9)}{lim_{x \to - 3} (9 - x^{2}) (x + 3)}$

Step 2.1.2.2.3.2

Multiply $- 1$ by $9$ .

$\frac{0 - (9 - 9)}{lim_{x \to - 3} (9 - x^{2}) (x + 3)}$

Step 2.1.2.2.4

Subtract $9$ from $9$ .

$\frac{0 - 0}{lim_{x \to - 3} (9 - x^{2}) (x + 3)}$

Step 2.1.2.2.5

Multiply $- 1$ by $0$ .

$\frac{0 + 0}{lim_{x \to - 3} (9 - x^{2}) (x + 3)}$

Step 2.1.2.3

Add $0$ and $0$ .

$\frac{0}{lim_{x \to - 3} (9 - x^{2}) (x + 3)}$

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 $- 3$ .

$\frac{0}{lim_{x \to - 3} 9 - x^{2} \cdot lim_{x \to - 3} x + 3}$

Step 2.1.3.2

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $- 3$ .

$\frac{0}{(lim_{x \to - 3} 9 - lim_{x \to - 3} x^{2}) \cdot lim_{x \to - 3} x + 3}$

Step 2.1.3.3

Evaluate the limit of $9$ which is constant as $x$ approaches $- 3$ .

$\frac{0}{(9 - lim_{x \to - 3} x^{2}) \cdot lim_{x \to - 3} x + 3}$

Step 2.1.3.4

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

$\frac{0}{(9 - {(lim_{x \to - 3} x)}^{2}) \cdot lim_{x \to - 3} x + 3}$

Step 2.1.3.5

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $- 3$ .

$\frac{0}{(9 - {(lim_{x \to - 3} x)}^{2}) \cdot (lim_{x \to - 3} x + lim_{x \to - 3} 3)}$

Step 2.1.3.6

Evaluate the limit of $3$ which is constant as $x$ approaches $- 3$ .

$\frac{0}{(9 - {(lim_{x \to - 3} x)}^{2}) \cdot (lim_{x \to - 3} x + 3)}$

Step 2.1.3.7

Evaluate the limits by plugging in $- 3$ for all occurrences of $x$ .

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Step 2.1.3.7.1

Evaluate the limit of $x$ by plugging in $- 3$ for $x$ .

$\frac{0}{(9 - {(- 3)}^{2}) \cdot (lim_{x \to - 3} x + 3)}$

Step 2.1.3.7.2

Evaluate the limit of $x$ by plugging in $- 3$ for $x$ .

$\frac{0}{(9 - {(- 3)}^{2}) \cdot (- 3 + 3)}$

Step 2.1.3.8

Simplify the answer.

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Step 2.1.3.8.1

Simplify each term.

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Step 2.1.3.8.1.1

Raise $- 3$ to the power of $2$ .

$\frac{0}{(9 - 1 \cdot 9) \cdot (- 3 + 3)}$

Step 2.1.3.8.1.2

Multiply $- 1$ by $9$ .

$\frac{0}{(9 - 9) \cdot (- 3 + 3)}$

Step 2.1.3.8.2

Subtract $9$ from $9$ .

$\frac{0}{0 \cdot (- 3 + 3)}$

Step 2.1.3.8.3

Add $- 3$ and $3$ .

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

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.

$lim_{x \to - 3} \frac{6 (x + 3) - (9 - x^{2})}{(9 - x^{2}) (x + 3)} = lim_{x \to - 3} \frac{\frac{d}{d x} [6 (x + 3) - (9 - x^{2})]}{\frac{d}{d x} [(9 - x^{2}) (x + 3)]}$

Step 2.3

Find the derivative of the numerator and denominator.

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Step 2.3.1

Differentiate the numerator and denominator.

$lim_{x \to - 3} \frac{\frac{d}{d x} [6 (x + 3) - (9 - x^{2})]}{\frac{d}{d x} [(9 - x^{2}) (x + 3)]}$

Step 2.3.2

By the Sum Rule, the derivative of $6 (x + 3) - (9 - x^{2})$ with respect to $x$ is $\frac{d}{d x} [6 (x + 3)] + \frac{d}{d x} [- (9 - x^{2})]$ .

$lim_{x \to - 3} \frac{\frac{d}{d x} [6 (x + 3)] + \frac{d}{d x} [- (9 - x^{2})]}{\frac{d}{d x} [(9 - x^{2}) (x + 3)]}$

Step 2.3.3

Evaluate $\frac{d}{d x} [6 (x + 3)]$ .

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Step 2.3.3.1

Since $6$ is constant with respect to $x$ , the derivative of $6 (x + 3)$ with respect to $x$ is $6 \frac{d}{d x} [x + 3]$ .

$lim_{x \to - 3} \frac{6 \frac{d}{d x} [x + 3] + \frac{d}{d x} [- (9 - x^{2})]}{\frac{d}{d x} [(9 - x^{2}) (x + 3)]}$

Step 2.3.3.2

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

$lim_{x \to - 3} \frac{6 (\frac{d}{d x} [x] + \frac{d}{d x} [3]) + \frac{d}{d x} [- (9 - x^{2})]}{\frac{d}{d x} [(9 - x^{2}) (x + 3)]}$

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 $n = 1$ .

$lim_{x \to - 3} \frac{6 (1 + \frac{d}{d x} [3]) + \frac{d}{d x} [- (9 - x^{2})]}{\frac{d}{d x} [(9 - x^{2}) (x + 3)]}$

Step 2.3.3.4

Since $3$ is constant with respect to $x$ , the derivative of $3$ with respect to $x$ is $0$ .

$lim_{x \to - 3} \frac{6 (1 + 0) + \frac{d}{d x} [- (9 - x^{2})]}{\frac{d}{d x} [(9 - x^{2}) (x + 3)]}$

Step 2.3.3.5

Add $1$ and $0$ .

$lim_{x \to - 3} \frac{6 \cdot 1 + \frac{d}{d x} [- (9 - x^{2})]}{\frac{d}{d x} [(9 - x^{2}) (x + 3)]}$

Step 2.3.3.6

Multiply $6$ by $1$ .

$lim_{x \to - 3} \frac{6 + \frac{d}{d x} [- (9 - x^{2})]}{\frac{d}{d x} [(9 - x^{2}) (x + 3)]}$

Step 2.3.4

Evaluate $\frac{d}{d x} [- (9 - x^{2})]$ .

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Step 2.3.4.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- (9 - x^{2})$ with respect to $x$ is $- \frac{d}{d x} [9 - x^{2}]$ .

$lim_{x \to - 3} \frac{6 - \frac{d}{d x} [9 - x^{2}]}{\frac{d}{d x} [(9 - x^{2}) (x + 3)]}$

Step 2.3.4.2

By the Sum Rule, the derivative of $9 - x^{2}$ with respect to $x$ is $\frac{d}{d x} [9] + \frac{d}{d x} [- x^{2}]$ .

$lim_{x \to - 3} \frac{6 - (\frac{d}{d x} [9] + \frac{d}{d x} [- x^{2}])}{\frac{d}{d x} [(9 - x^{2}) (x + 3)]}$

Step 2.3.4.3

Since $9$ is constant with respect to $x$ , the derivative of $9$ with respect to $x$ is $0$ .

$lim_{x \to - 3} \frac{6 - (0 + \frac{d}{d x} [- x^{2}])}{\frac{d}{d x} [(9 - x^{2}) (x + 3)]}$

Step 2.3.4.4

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

$lim_{x \to - 3} \frac{6 - (0 - \frac{d}{d x} [x^{2}])}{\frac{d}{d x} [(9 - x^{2}) (x + 3)]}$

Step 2.3.4.5

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

$lim_{x \to - 3} \frac{6 - (0 - (2 x))}{\frac{d}{d x} [(9 - x^{2}) (x + 3)]}$

Step 2.3.4.6

Multiply $2$ by $- 1$ .

$lim_{x \to - 3} \frac{6 - (0 - 2 x)}{\frac{d}{d x} [(9 - x^{2}) (x + 3)]}$

Step 2.3.4.7

Subtract $2 x$ from $0$ .

$lim_{x \to - 3} \frac{6 - (- 2 x)}{\frac{d}{d x} [(9 - x^{2}) (x + 3)]}$

Step 2.3.4.8

Multiply $- 2$ by $- 1$ .

$lim_{x \to - 3} \frac{6 + 2 x}{\frac{d}{d x} [(9 - x^{2}) (x + 3)]}$

Step 2.3.5

Reorder terms.

$lim_{x \to - 3} \frac{2 x + 6}{\frac{d}{d x} [(9 - x^{2}) (x + 3)]}$

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 $f (x) = 9 - x^{2}$ and $g (x) = x + 3$ .

$lim_{x \to - 3} \frac{2 x + 6}{(9 - x^{2}) \frac{d}{d x} [x + 3] + (x + 3) \frac{d}{d x} [9 - x^{2}]}$

Step 2.3.7

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

$lim_{x \to - 3} \frac{2 x + 6}{(9 - x^{2}) (\frac{d}{d x} [x] + \frac{d}{d x} [3]) + (x + 3) \frac{d}{d x} [9 - x^{2}]}$

Step 2.3.8

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

$lim_{x \to - 3} \frac{2 x + 6}{(9 - x^{2}) (1 + \frac{d}{d x} [3]) + (x + 3) \frac{d}{d x} [9 - x^{2}]}$

Step 2.3.9

Since $3$ is constant with respect to $x$ , the derivative of $3$ with respect to $x$ is $0$ .

$lim_{x \to - 3} \frac{2 x + 6}{(9 - x^{2}) (1 + 0) + (x + 3) \frac{d}{d x} [9 - x^{2}]}$

Step 2.3.10

Add $1$ and $0$ .

$lim_{x \to - 3} \frac{2 x + 6}{(9 - x^{2}) \cdot 1 + (x + 3) \frac{d}{d x} [9 - x^{2}]}$

Step 2.3.11

Multiply $9 - x^{2}$ by $1$ .

$lim_{x \to - 3} \frac{2 x + 6}{9 - x^{2} + (x + 3) \frac{d}{d x} [9 - x^{2}]}$

Step 2.3.12

By the Sum Rule, the derivative of $9 - x^{2}$ with respect to $x$ is $\frac{d}{d x} [9] + \frac{d}{d x} [- x^{2}]$ .

$lim_{x \to - 3} \frac{2 x + 6}{9 - x^{2} + (x + 3) (\frac{d}{d x} [9] + \frac{d}{d x} [- x^{2}])}$

Step 2.3.13

Since $9$ is constant with respect to $x$ , the derivative of $9$ with respect to $x$ is $0$ .

$lim_{x \to - 3} \frac{2 x + 6}{9 - x^{2} + (x + 3) (0 + \frac{d}{d x} [- x^{2}])}$

Step 2.3.14

Add $0$ and $\frac{d}{d x} [- x^{2}]$ .

$lim_{x \to - 3} \frac{2 x + 6}{9 - x^{2} + (x + 3) \frac{d}{d x} [- x^{2}]}$

Step 2.3.15

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

$lim_{x \to - 3} \frac{2 x + 6}{9 - x^{2} + (x + 3) (- \frac{d}{d x} [x^{2}])}$

Step 2.3.16

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

$lim_{x \to - 3} \frac{2 x + 6}{9 - x^{2} + (x + 3) (- (2 x))}$

Step 2.3.17

Multiply $2$ by $- 1$ .

$lim_{x \to - 3} \frac{2 x + 6}{9 - x^{2} + (x + 3) (- 2 x)}$

Step 2.3.18

Simplify.

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Step 2.3.18.1

Apply the distributive property.

$lim_{x \to - 3} \frac{2 x + 6}{9 - x^{2} + x (- 2 x) + 3 (- 2 x)}$

Step 2.3.18.2

Combine terms.

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Step 2.3.18.2.1

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

$lim_{x \to - 3} \frac{2 x + 6}{9 - x^{2} - 2 (x^{1} x) + 3 (- 2 x)}$

Step 2.3.18.2.2

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

$lim_{x \to - 3} \frac{2 x + 6}{9 - x^{2} - 2 (x^{1} x^{1}) + 3 (- 2 x)}$

Step 2.3.18.2.3

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

$lim_{x \to - 3} \frac{2 x + 6}{9 - x^{2} - 2 x^{1 + 1} + 3 (- 2 x)}$

Step 2.3.18.2.4

Add $1$ and $1$ .

$lim_{x \to - 3} \frac{2 x + 6}{9 - x^{2} - 2 x^{2} + 3 (- 2 x)}$

Step 2.3.18.2.5

Multiply $- 2$ by $3$ .

$lim_{x \to - 3} \frac{2 x + 6}{9 - x^{2} - 2 x^{2} - 6 x}$

Step 2.3.18.2.6

Subtract $2 x^{2}$ from $- x^{2}$ .

$lim_{x \to - 3} \frac{2 x + 6}{9 - 3 x^{2} - 6 x}$

Step 2.3.18.3

Reorder terms.

$lim_{x \to - 3} \frac{2 x + 6}{- 3 x^{2} - 6 x + 9}$

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.

$\frac{lim_{x \to - 3} 2 x + 6}{lim_{x \to - 3} - 3 x^{2} - 6 x + 9}$

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 $- 3$ .

$\frac{lim_{x \to - 3} 2 x + lim_{x \to - 3} 6}{lim_{x \to - 3} - 3 x^{2} - 6 x + 9}$

Step 3.1.2.1.2

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

$\frac{2 lim_{x \to - 3} x + lim_{x \to - 3} 6}{lim_{x \to - 3} - 3 x^{2} - 6 x + 9}$

Step 3.1.2.1.3

Evaluate the limit of $6$ which is constant as $x$ approaches $- 3$ .

$\frac{2 lim_{x \to - 3} x + 6}{lim_{x \to - 3} - 3 x^{2} - 6 x + 9}$

Step 3.1.2.2

Evaluate the limit of $x$ by plugging in $- 3$ for $x$ .

$\frac{2 \cdot - 3 + 6}{lim_{x \to - 3} - 3 x^{2} - 6 x + 9}$

Step 3.1.2.3

Simplify the answer.

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Step 3.1.2.3.1

Multiply $2$ by $- 3$ .

$\frac{- 6 + 6}{lim_{x \to - 3} - 3 x^{2} - 6 x + 9}$

Step 3.1.2.3.2

Add $- 6$ and $6$ .

$\frac{0}{lim_{x \to - 3} - 3 x^{2} - 6 x + 9}$

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 $- 3$ .

$\frac{0}{- lim_{x \to - 3} 3 x^{2} - lim_{x \to - 3} 6 x + lim_{x \to - 3} 9}$

Step 3.1.3.2

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

$\frac{0}{- 3 lim_{x \to - 3} x^{2} - lim_{x \to - 3} 6 x + lim_{x \to - 3} 9}$

Step 3.1.3.3

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

$\frac{0}{- 3 {(lim_{x \to - 3} x)}^{2} - lim_{x \to - 3} 6 x + lim_{x \to - 3} 9}$

Step 3.1.3.4

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

$\frac{0}{- 3 {(lim_{x \to - 3} x)}^{2} - 6 lim_{x \to - 3} x + lim_{x \to - 3} 9}$

Step 3.1.3.5

Evaluate the limit of $9$ which is constant as $x$ approaches $- 3$ .

$\frac{0}{- 3 {(lim_{x \to - 3} x)}^{2} - 6 lim_{x \to - 3} x + 9}$

Step 3.1.3.6

Evaluate the limits by plugging in $- 3$ for all occurrences of $x$ .

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Step 3.1.3.6.1

Evaluate the limit of $x$ by plugging in $- 3$ for $x$ .

$\frac{0}{- 3 {(- 3)}^{2} - 6 lim_{x \to - 3} x + 9}$

Step 3.1.3.6.2

Evaluate the limit of $x$ by plugging in $- 3$ for $x$ .

$\frac{0}{- 3 {(- 3)}^{2} - 6 \cdot - 3 + 9}$

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

Multiply $- 3$ by ${(- 3)}^{2}$ by adding the exponents.

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Step 3.1.3.7.1.1.1

Multiply $- 3$ by ${(- 3)}^{2}$ .

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Step 3.1.3.7.1.1.1.1

Raise $- 3$ to the power of $1$ .

$\frac{0}{{(- 3)}^{1} {(- 3)}^{2} - 6 \cdot - 3 + 9}$

Step 3.1.3.7.1.1.1.2

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

$\frac{0}{{(- 3)}^{1 + 2} - 6 \cdot - 3 + 9}$

Step 3.1.3.7.1.1.2

Add $1$ and $2$ .

$\frac{0}{{(- 3)}^{3} - 6 \cdot - 3 + 9}$

Step 3.1.3.7.1.2

Raise $- 3$ to the power of $3$ .

$\frac{0}{- 27 - 6 \cdot - 3 + 9}$

Step 3.1.3.7.1.3

Multiply $- 6$ by $- 3$ .

$\frac{0}{- 27 + 18 + 9}$

Step 3.1.3.7.2

Add $- 27$ and $18$ .

$\frac{0}{- 9 + 9}$

Step 3.1.3.7.3

Add $- 9$ and $9$ .

$\frac{0}{0}$

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.

$lim_{x \to - 3} \frac{2 x + 6}{- 3 x^{2} - 6 x + 9} = lim_{x \to - 3} \frac{\frac{d}{d x} [2 x + 6]}{\frac{d}{d x} [- 3 x^{2} - 6 x + 9]}$

Step 3.3

Find the derivative of the numerator and denominator.

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Step 3.3.1

Differentiate the numerator and denominator.

$lim_{x \to - 3} \frac{\frac{d}{d x} [2 x + 6]}{\frac{d}{d x} [- 3 x^{2} - 6 x + 9]}$

Step 3.3.2

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

$lim_{x \to - 3} \frac{\frac{d}{d x} [2 x] + \frac{d}{d x} [6]}{\frac{d}{d x} [- 3 x^{2} - 6 x + 9]}$

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

$lim_{x \to - 3} \frac{2 \frac{d}{d x} [x] + \frac{d}{d x} [6]}{\frac{d}{d x} [- 3 x^{2} - 6 x + 9]}$

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 $n = 1$ .

$lim_{x \to - 3} \frac{2 \cdot 1 + \frac{d}{d x} [6]}{\frac{d}{d x} [- 3 x^{2} - 6 x + 9]}$

Step 3.3.3.3

Multiply $2$ by $1$ .

$lim_{x \to - 3} \frac{2 + \frac{d}{d x} [6]}{\frac{d}{d x} [- 3 x^{2} - 6 x + 9]}$

Step 3.3.4

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

$lim_{x \to - 3} \frac{2 + 0}{\frac{d}{d x} [- 3 x^{2} - 6 x + 9]}$

Step 3.3.5

Add $2$ and $0$ .

$lim_{x \to - 3} \frac{2}{\frac{d}{d x} [- 3 x^{2} - 6 x + 9]}$

Step 3.3.6

By the Sum Rule, the derivative of $- 3 x^{2} - 6 x + 9$ with respect to $x$ is $\frac{d}{d x} [- 3 x^{2}] + \frac{d}{d x} [- 6 x] + \frac{d}{d x} [9]$ .

$lim_{x \to - 3} \frac{2}{\frac{d}{d x} [- 3 x^{2}] + \frac{d}{d x} [- 6 x] + \frac{d}{d x} [9]}$

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

$lim_{x \to - 3} \frac{2}{- 3 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 6 x] + \frac{d}{d x} [9]}$

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 $n = 2$ .

$lim_{x \to - 3} \frac{2}{- 3 (2 x) + \frac{d}{d x} [- 6 x] + \frac{d}{d x} [9]}$

Step 3.3.7.3

Multiply $2$ by $- 3$ .

$lim_{x \to - 3} \frac{2}{- 6 x + \frac{d}{d x} [- 6 x] + \frac{d}{d x} [9]}$

Step 3.3.8

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

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Step 3.3.8.1

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

$lim_{x \to - 3} \frac{2}{- 6 x - 6 \frac{d}{d x} [x] + \frac{d}{d x} [9]}$

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 $n = 1$ .

$lim_{x \to - 3} \frac{2}{- 6 x - 6 \cdot 1 + \frac{d}{d x} [9]}$

Step 3.3.8.3

Multiply $- 6$ by $1$ .

$lim_{x \to - 3} \frac{2}{- 6 x - 6 + \frac{d}{d x} [9]}$

Step 3.3.9

Since $9$ is constant with respect to $x$ , the derivative of $9$ with respect to $x$ is $0$ .

$lim_{x \to - 3} \frac{2}{- 6 x - 6 + 0}$

Step 3.3.10

Add $- 6 x - 6$ and $0$ .

$lim_{x \to - 3} \frac{2}{- 6 x - 6}$

Step 3.4

Cancel the common factor of $2$ and $- 6 x - 6$ .

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Step 3.4.1

Factor $2$ out of $2$ .

$lim_{x \to - 3} \frac{2 (1)}{- 6 x - 6}$

Step 3.4.2

Cancel the common factors.

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Step 3.4.2.1

Factor $2$ out of $- 6 x$ .

$lim_{x \to - 3} \frac{2 (1)}{2 (- 3 x) - 6}$

Step 3.4.2.2

Factor $2$ out of $- 6$ .

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

Step 3.4.2.3

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

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

Step 3.4.2.4

Cancel the common factor.

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

Step 3.4.2.5

Rewrite the expression.

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

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 $- 3$ .

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

Step 4.2

Evaluate the limit of $1$ which is constant as $x$ approaches $- 3$ .

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

Step 4.3

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $- 3$ .

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

Step 4.4

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

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

Step 4.5

Evaluate the limit of $3$ which is constant as $x$ approaches $- 3$ .

$\frac{1}{- 3 lim_{x \to - 3} x - 1 \cdot 3}$

Step 5

Evaluate the limit of $x$ by plugging in $- 3$ for $x$ .

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

Step 6

Simplify the denominator.

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Step 6.1

Multiply $- 3$ by $- 3$ .

$\frac{1}{9 - 1 \cdot 3}$

Step 6.2

Multiply $- 1$ by $3$ .

$\frac{1}{9 - 3}$

Step 6.3

Subtract $3$ from $9$ .

$\frac{1}{6}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$\frac{1}{6}$

Decimal Form:

$0.1\overline{6}$