Calculus Examples

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

Step 1

Apply L'Hospital's rule.

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Step 1.1

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

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Step 1.1.1

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

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

Step 1.1.2

Evaluate the limit of the numerator.

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Step 1.1.2.1

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $- 1$ .

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

Step 1.1.2.2

Move the exponent $3$ from $x^{3}$ outside the limit using the Limits Power Rule.

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

Step 1.1.2.3

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

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

Step 1.1.2.4

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

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

Step 1.1.2.5

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

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

Step 1.1.2.6

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

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

Step 1.1.2.7

Evaluate the limits by plugging in $- 1$ for all occurrences of $x$ .

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Step 1.1.2.7.1

Evaluate the limit of $x$ by plugging in $- 1$ for $x$ .

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

Step 1.1.2.7.2

Evaluate the limit of $x$ by plugging in $- 1$ for $x$ .

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

Step 1.1.2.7.3

Evaluate the limit of $x$ by plugging in $- 1$ for $x$ .

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

Step 1.1.2.8

Simplify the answer.

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Step 1.1.2.8.1

Simplify each term.

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Step 1.1.2.8.1.1

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

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

Step 1.1.2.8.1.2

Raise $- 1$ to the power of $2$ .

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

Step 1.1.2.8.1.3

Multiply $3$ by $1$ .

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

Step 1.1.2.8.1.4

Multiply $3$ by $- 1$ .

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

Step 1.1.2.8.2

Add $- 1$ and $3$ .

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

Step 1.1.2.8.3

Subtract $3$ from $2$ .

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

Step 1.1.2.8.4

Add $- 1$ and $1$ .

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

Step 1.1.3

Evaluate the limit of the denominator.

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Step 1.1.3.1

Evaluate the limit.

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Step 1.1.3.1.1

Move the exponent $2$ from ${(x + 1)}^{2}$ outside the limit using the Limits Power Rule.

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

Step 1.1.3.1.2

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $- 1$ .

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

Step 1.1.3.1.3

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

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

Step 1.1.3.2

Evaluate the limit of $x$ by plugging in $- 1$ for $x$ .

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

Step 1.1.3.3

Simplify the answer.

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Step 1.1.3.3.1

Add $- 1$ and $1$ .

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

Step 1.1.3.3.2

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

$\frac{0}{0}$

Step 1.1.3.3.3

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

Undefined

$\frac{0}{0}$

Step 1.1.3.4

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

Undefined

$\frac{0}{0}$

Step 1.1.4

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

Undefined

$\frac{0}{0}$

Step 1.2

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.

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

Step 1.3

Find the derivative of the numerator and denominator.

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Step 1.3.1

Differentiate the numerator and denominator.

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

Step 1.3.2

By the Sum Rule, the derivative of $x^{3} + 3 x^{2} + 3 x + 1$ with respect to $x$ is $\frac{d}{d x} [x^{3}] + \frac{d}{d x} [3 x^{2}] + \frac{d}{d x} [3 x] + \frac{d}{d x} [1]$ .

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

Step 1.3.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

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

Step 1.3.4

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

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Step 1.3.4.1

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

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

Step 1.3.4.2

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

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

Step 1.3.4.3

Multiply $2$ by $3$ .

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

Step 1.3.5

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

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Step 1.3.5.1

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

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

Step 1.3.5.2

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

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

Step 1.3.5.3

Multiply $3$ by $1$ .

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

Step 1.3.6

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

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

Step 1.3.7

Add $3 x^{2} + 6 x + 3$ and $0$ .

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

Step 1.3.8

Rewrite ${(x + 1)}^{2}$ as $(x + 1) (x + 1)$ .

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

Step 1.3.9

Expand $(x + 1) (x + 1)$ using the FOIL Method.

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Step 1.3.9.1

Apply the distributive property.

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

Step 1.3.9.2

Apply the distributive property.

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

Step 1.3.9.3

Apply the distributive property.

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

Step 1.3.10

Simplify and combine like terms.

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Step 1.3.10.1

Simplify each term.

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Step 1.3.10.1.1

Multiply $x$ by $x$ .

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

Step 1.3.10.1.2

Multiply $x$ by $1$ .

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

Step 1.3.10.1.3

Multiply $x$ by $1$ .

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

Step 1.3.10.1.4

Multiply $1$ by $1$ .

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

Step 1.3.10.2

Add $x$ and $x$ .

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

Step 1.3.11

By the Sum Rule, the derivative of $x^{2} + 2 x + 1$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [2 x] + \frac{d}{d x} [1]$ .

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

Step 1.3.12

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

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

Step 1.3.13

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

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

Step 1.3.14

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

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

Step 1.3.15

Multiply $2$ by $1$ .

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

Step 1.3.16

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

$lim_{x \to - 1} \frac{3 x^{2} + 6 x + 3}{2 x + 2 + 0}$

Step 1.3.17

Add $2 x + 2$ and $0$ .

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

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.

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

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 $- 1$ .

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

Step 2.1.2.2

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

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

Step 2.1.2.3

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

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

Step 2.1.2.4

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

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

Step 2.1.2.5

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

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

Step 2.1.2.6

Evaluate the limits by plugging in $- 1$ for all occurrences of $x$ .

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Step 2.1.2.6.1

Evaluate the limit of $x$ by plugging in $- 1$ for $x$ .

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

Step 2.1.2.6.2

Evaluate the limit of $x$ by plugging in $- 1$ for $x$ .

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

Step 2.1.2.7

Simplify the answer.

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Step 2.1.2.7.1

Simplify each term.

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Step 2.1.2.7.1.1

Raise $- 1$ to the power of $2$ .

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

Step 2.1.2.7.1.2

Multiply $3$ by $1$ .

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

Step 2.1.2.7.1.3

Multiply $6$ by $- 1$ .

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

Step 2.1.2.7.2

Subtract $6$ from $3$ .

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

Step 2.1.2.7.3

Add $- 3$ and $3$ .

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

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 $- 1$ .

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

Step 2.1.3.1.2

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

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

Step 2.1.3.1.3

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

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

Step 2.1.3.2

Evaluate the limit of $x$ by plugging in $- 1$ for $x$ .

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

Step 2.1.3.3

Simplify the answer.

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Step 2.1.3.3.1

Multiply $2$ by $- 1$ .

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

Step 2.1.3.3.2

Add $- 2$ and $2$ .

$\frac{0}{0}$

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.

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

Step 2.3

Find the derivative of the numerator and denominator.

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Step 2.3.1

Differentiate the numerator and denominator.

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

Step 2.3.2

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

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

Step 2.3.3

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

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Step 2.3.3.1

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

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

Step 2.3.3.2

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

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

Step 2.3.3.3

Multiply $2$ by $3$ .

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

Step 2.3.4

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

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Step 2.3.4.1

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

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

Step 2.3.4.2

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

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

Step 2.3.4.3

Multiply $6$ by $1$ .

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

Step 2.3.5

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

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

Step 2.3.6

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

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

Step 2.3.7

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

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

Step 2.3.8

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

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Step 2.3.8.1

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

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

Step 2.3.8.2

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

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

Step 2.3.8.3

Multiply $2$ by $1$ .

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

Step 2.3.9

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

$lim_{x \to - 1} \frac{6 x + 6}{2 + 0}$

Step 2.3.10

Add $2$ and $0$ .

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

Step 2.4

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

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Step 2.4.1

Factor $2$ out of $6 x$ .

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

Step 2.4.2

Factor $2$ out of $6$ .

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

Step 2.4.3

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

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

Step 2.4.4

Cancel the common factors.

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Step 2.4.4.1

Factor $2$ out of $2$ .

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

Step 2.4.4.2

Cancel the common factor.

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

Step 2.4.4.3

Rewrite the expression.

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

Step 2.4.4.4

Divide $3 x + 3$ by $1$ .

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

Step 3

Evaluate the limit.

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Step 3.1

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $- 1$ .

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

Step 3.2

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

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

Step 3.3

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

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

Step 4

Evaluate the limit of $x$ by plugging in $- 1$ for $x$ .

$3 \cdot - 1 + 3$

Step 5

Simplify the answer.

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Step 5.1

Multiply $3$ by $- 1$ .

$- 3 + 3$

Step 5.2

Add $- 3$ and $3$ .

$0$