Calculus Examples

$lim_{x \to - 4} \frac{3 {(x + 1)}^{2} - 27}{x + 4}$

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.

$\frac{lim_{x \to - 4} 3 {(x + 1)}^{2} - 27}{lim_{x \to - 4} x + 4}$

Step 1.1.2

Evaluate the limit of the numerator.

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Step 1.1.2.1

Evaluate the limit.

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Step 1.1.2.1.1

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $- 4$ .

$\frac{lim_{x \to - 4} 3 {(x + 1)}^{2} - lim_{x \to - 4} 27}{lim_{x \to - 4} x + 4}$

Step 1.1.2.1.2

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

$\frac{3 lim_{x \to - 4} {(x + 1)}^{2} - lim_{x \to - 4} 27}{lim_{x \to - 4} x + 4}$

Step 1.1.2.1.3

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

$\frac{3 {(lim_{x \to - 4} x + 1)}^{2} - lim_{x \to - 4} 27}{lim_{x \to - 4} x + 4}$

Step 1.1.2.1.4

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $- 4$ .

$\frac{3 {(lim_{x \to - 4} x + lim_{x \to - 4} 1)}^{2} - lim_{x \to - 4} 27}{lim_{x \to - 4} x + 4}$

Step 1.1.2.1.5

Evaluate the limit of $1$ which is constant as $x$ approaches $- 4$ .

$\frac{3 {(lim_{x \to - 4} x + 1)}^{2} - lim_{x \to - 4} 27}{lim_{x \to - 4} x + 4}$

Step 1.1.2.1.6

Evaluate the limit of $27$ which is constant as $x$ approaches $- 4$ .

$\frac{3 {(lim_{x \to - 4} x + 1)}^{2} - 1 \cdot 27}{lim_{x \to - 4} x + 4}$

Step 1.1.2.2

Evaluate the limit of $x$ by plugging in $- 4$ for $x$ .

$\frac{3 {(- 4 + 1)}^{2} - 1 \cdot 27}{lim_{x \to - 4} x + 4}$

Step 1.1.2.3

Simplify the answer.

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Step 1.1.2.3.1

Simplify each term.

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Step 1.1.2.3.1.1

Add $- 4$ and $1$ .

$\frac{3 {(- 3)}^{2} - 1 \cdot 27}{lim_{x \to - 4} x + 4}$

Step 1.1.2.3.1.2

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

$\frac{3 \cdot 9 - 1 \cdot 27}{lim_{x \to - 4} x + 4}$

Step 1.1.2.3.1.3

Multiply $3$ by $9$ .

$\frac{27 - 1 \cdot 27}{lim_{x \to - 4} x + 4}$

Step 1.1.2.3.1.4

Multiply $- 1$ by $27$ .

$\frac{27 - 27}{lim_{x \to - 4} x + 4}$

Step 1.1.2.3.2

Subtract $27$ from $27$ .

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

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

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $- 4$ .

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

Step 1.1.3.1.2

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

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

Step 1.1.3.2

Evaluate the limit of $x$ by plugging in $- 4$ for $x$ .

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

Step 1.1.3.3

Add $- 4$ and $4$ .

$\frac{0}{0}$

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.

$lim_{x \to - 4} \frac{3 {(x + 1)}^{2} - 27}{x + 4} = lim_{x \to - 4} \frac{\frac{d}{d x} [3 {(x + 1)}^{2} - 27]}{\frac{d}{d x} [x + 4]}$

Step 1.3

Find the derivative of the numerator and denominator.

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Step 1.3.1

Differentiate the numerator and denominator.

$lim_{x \to - 4} \frac{\frac{d}{d x} [3 {(x + 1)}^{2} - 27]}{\frac{d}{d x} [x + 4]}$

Step 1.3.2

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

$lim_{x \to - 4} \frac{\frac{d}{d x} [3 ((x + 1) (x + 1)) - 27]}{\frac{d}{d x} [x + 4]}$

Step 1.3.3

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

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Step 1.3.3.1

Apply the distributive property.

$lim_{x \to - 4} \frac{\frac{d}{d x} [3 (x (x + 1) + 1 (x + 1)) - 27]}{\frac{d}{d x} [x + 4]}$

Step 1.3.3.2

Apply the distributive property.

$lim_{x \to - 4} \frac{\frac{d}{d x} [3 (x \cdot x + x \cdot 1 + 1 (x + 1)) - 27]}{\frac{d}{d x} [x + 4]}$

Step 1.3.3.3

Apply the distributive property.

$lim_{x \to - 4} \frac{\frac{d}{d x} [3 (x \cdot x + x \cdot 1 + 1 x + 1 \cdot 1) - 27]}{\frac{d}{d x} [x + 4]}$

Step 1.3.4

Simplify and combine like terms.

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Step 1.3.4.1

Simplify each term.

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Step 1.3.4.1.1

Multiply $x$ by $x$ .

$lim_{x \to - 4} \frac{\frac{d}{d x} [3 (x^{2} + x \cdot 1 + 1 x + 1 \cdot 1) - 27]}{\frac{d}{d x} [x + 4]}$

Step 1.3.4.1.2

Multiply $x$ by $1$ .

$lim_{x \to - 4} \frac{\frac{d}{d x} [3 (x^{2} + x + 1 x + 1 \cdot 1) - 27]}{\frac{d}{d x} [x + 4]}$

Step 1.3.4.1.3

Multiply $x$ by $1$ .

$lim_{x \to - 4} \frac{\frac{d}{d x} [3 (x^{2} + x + x + 1 \cdot 1) - 27]}{\frac{d}{d x} [x + 4]}$

Step 1.3.4.1.4

Multiply $1$ by $1$ .

$lim_{x \to - 4} \frac{\frac{d}{d x} [3 (x^{2} + x + x + 1) - 27]}{\frac{d}{d x} [x + 4]}$

Step 1.3.4.2

Add $x$ and $x$ .

$lim_{x \to - 4} \frac{\frac{d}{d x} [3 (x^{2} + 2 x + 1) - 27]}{\frac{d}{d x} [x + 4]}$

Step 1.3.5

By the Sum Rule, the derivative of $3 (x^{2} + 2 x + 1) - 27$ with respect to $x$ is $\frac{d}{d x} [3 (x^{2} + 2 x + 1)] + \frac{d}{d x} [- 27]$ .

$lim_{x \to - 4} \frac{\frac{d}{d x} [3 (x^{2} + 2 x + 1)] + \frac{d}{d x} [- 27]}{\frac{d}{d x} [x + 4]}$

Step 1.3.6

Evaluate $\frac{d}{d x} [3 (x^{2} + 2 x + 1)]$ .

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Step 1.3.6.1

Since $3$ is constant with respect to $x$ , the derivative of $3 (x^{2} + 2 x + 1)$ with respect to $x$ is $3 \frac{d}{d x} [x^{2} + 2 x + 1]$ .

$lim_{x \to - 4} \frac{3 \frac{d}{d x} [x^{2} + 2 x + 1] + \frac{d}{d x} [- 27]}{\frac{d}{d x} [x + 4]}$

Step 1.3.6.2

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

$lim_{x \to - 4} \frac{3 (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [2 x] + \frac{d}{d x} [1]) + \frac{d}{d x} [- 27]}{\frac{d}{d x} [x + 4]}$

Step 1.3.6.3

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

$lim_{x \to - 4} \frac{3 (2 x + \frac{d}{d x} [2 x] + \frac{d}{d x} [1]) + \frac{d}{d x} [- 27]}{\frac{d}{d x} [x + 4]}$

Step 1.3.6.4

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

$lim_{x \to - 4} \frac{3 (2 x + 2 \frac{d}{d x} [x] + \frac{d}{d x} [1]) + \frac{d}{d x} [- 27]}{\frac{d}{d x} [x + 4]}$

Step 1.3.6.5

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

$lim_{x \to - 4} \frac{3 (2 x + 2 \cdot 1 + \frac{d}{d x} [1]) + \frac{d}{d x} [- 27]}{\frac{d}{d x} [x + 4]}$

Step 1.3.6.6

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

$lim_{x \to - 4} \frac{3 (2 x + 2 \cdot 1 + 0) + \frac{d}{d x} [- 27]}{\frac{d}{d x} [x + 4]}$

Step 1.3.6.7

Multiply $2$ by $1$ .

$lim_{x \to - 4} \frac{3 (2 x + 2 + 0) + \frac{d}{d x} [- 27]}{\frac{d}{d x} [x + 4]}$

Step 1.3.6.8

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

$lim_{x \to - 4} \frac{3 (2 x + 2) + \frac{d}{d x} [- 27]}{\frac{d}{d x} [x + 4]}$

Step 1.3.7

Since $- 27$ is constant with respect to $x$ , the derivative of $- 27$ with respect to $x$ is $0$ .

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

Step 1.3.8

Simplify.

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Step 1.3.8.1

Apply the distributive property.

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

Step 1.3.8.2

Combine terms.

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Step 1.3.8.2.1

Multiply $2$ by $3$ .

$lim_{x \to - 4} \frac{6 x + 3 \cdot 2 + 0}{\frac{d}{d x} [x + 4]}$

Step 1.3.8.2.2

Multiply $3$ by $2$ .

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

Step 1.3.8.2.3

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

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

Step 1.3.9

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

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

Step 1.3.10

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

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

Step 1.3.11

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

$lim_{x \to - 4} \frac{6 x + 6}{1 + 0}$

Step 1.3.12

Add $1$ and $0$ .

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

Step 1.4

Divide $6 x + 6$ by $1$ .

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

Step 2

Evaluate the limit.

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Step 2.1

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $- 4$ .

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

Step 2.2

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

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

Step 2.3

Evaluate the limit of $6$ which is constant as $x$ approaches $- 4$ .

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

Step 3

Evaluate the limit of $x$ by plugging in $- 4$ for $x$ .

$6 \cdot - 4 + 6$

Step 4

Simplify the answer.

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Step 4.1

Multiply $6$ by $- 4$ .

$- 24 + 6$

Step 4.2

Add $- 24$ and $6$ .

$- 18$