Calculus Examples

$lim_{x \to - 2} \frac{x^{3} + 5 x^{2} + 6 x}{x^{2} (x + 2) - (x + 2)}$

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.

$\frac{lim_{x \to - 2} x^{3} + 5 x^{2} + 6 x}{lim_{x \to - 2} x^{2} (x + 2) - (x + 2)}$

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 $- 2$ .

$\frac{lim_{x \to - 2} x^{3} + lim_{x \to - 2} 5 x^{2} + lim_{x \to - 2} 6 x}{lim_{x \to - 2} x^{2} (x + 2) - (x + 2)}$

Step 1.1.2.2

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

$\frac{{(lim_{x \to - 2} x)}^{3} + lim_{x \to - 2} 5 x^{2} + lim_{x \to - 2} 6 x}{lim_{x \to - 2} x^{2} (x + 2) - (x + 2)}$

Step 1.1.2.3

Move the term $5$ outside of the limit because it is constant with respect to $x$ .

$\frac{{(lim_{x \to - 2} x)}^{3} + 5 lim_{x \to - 2} x^{2} + lim_{x \to - 2} 6 x}{lim_{x \to - 2} x^{2} (x + 2) - (x + 2)}$

Step 1.1.2.4

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

$\frac{{(lim_{x \to - 2} x)}^{3} + 5 {(lim_{x \to - 2} x)}^{2} + lim_{x \to - 2} 6 x}{lim_{x \to - 2} x^{2} (x + 2) - (x + 2)}$

Step 1.1.2.5

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

$\frac{{(lim_{x \to - 2} x)}^{3} + 5 {(lim_{x \to - 2} x)}^{2} + 6 lim_{x \to - 2} x}{lim_{x \to - 2} x^{2} (x + 2) - (x + 2)}$

Step 1.1.2.6

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

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Step 1.1.2.6.1

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

$\frac{{(- 2)}^{3} + 5 {(lim_{x \to - 2} x)}^{2} + 6 lim_{x \to - 2} x}{lim_{x \to - 2} x^{2} (x + 2) - (x + 2)}$

Step 1.1.2.6.2

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

$\frac{{(- 2)}^{3} + 5 {(- 2)}^{2} + 6 lim_{x \to - 2} x}{lim_{x \to - 2} x^{2} (x + 2) - (x + 2)}$

Step 1.1.2.6.3

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

$\frac{{(- 2)}^{3} + 5 {(- 2)}^{2} + 6 \cdot - 2}{lim_{x \to - 2} x^{2} (x + 2) - (x + 2)}$

Step 1.1.2.7

Simplify the answer.

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Step 1.1.2.7.1

Simplify each term.

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Step 1.1.2.7.1.1

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

$\frac{- 8 + 5 {(- 2)}^{2} + 6 \cdot - 2}{lim_{x \to - 2} x^{2} (x + 2) - (x + 2)}$

Step 1.1.2.7.1.2

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

$\frac{- 8 + 5 \cdot 4 + 6 \cdot - 2}{lim_{x \to - 2} x^{2} (x + 2) - (x + 2)}$

Step 1.1.2.7.1.3

Multiply $5$ by $4$ .

$\frac{- 8 + 20 + 6 \cdot - 2}{lim_{x \to - 2} x^{2} (x + 2) - (x + 2)}$

Step 1.1.2.7.1.4

Multiply $6$ by $- 2$ .

$\frac{- 8 + 20 - 12}{lim_{x \to - 2} x^{2} (x + 2) - (x + 2)}$

Step 1.1.2.7.2

Add $- 8$ and $20$ .

$\frac{12 - 12}{lim_{x \to - 2} x^{2} (x + 2) - (x + 2)}$

Step 1.1.2.7.3

Subtract $12$ from $12$ .

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

Step 1.1.3

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

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Step 1.1.3.1

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

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

Step 1.1.3.2

Simplify each term.

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Step 1.1.3.2.1

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

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

Step 1.1.3.2.2

Add $- 2$ and $2$ .

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

Step 1.1.3.2.3

Multiply $4$ by $0$ .

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

Step 1.1.3.2.4

Add $- 2$ and $2$ .

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

Step 1.1.3.2.5

Multiply $- 1$ by $0$ .

$\frac{0}{0 + 0}$

Step 1.1.3.3

Add $0$ and $0$ .

$\frac{0}{0}$

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.

$lim_{x \to - 2} \frac{x^{3} + 5 x^{2} + 6 x}{x^{2} (x + 2) - (x + 2)} = lim_{x \to - 2} \frac{\frac{d}{d x} [x^{3} + 5 x^{2} + 6 x]}{\frac{d}{d x} [x^{2} (x + 2) - (x + 2)]}$

Step 1.3

Find the derivative of the numerator and denominator.

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Step 1.3.1

Differentiate the numerator and denominator.

$lim_{x \to - 2} \frac{\frac{d}{d x} [x^{3} + 5 x^{2} + 6 x]}{\frac{d}{d x} [x^{2} (x + 2) - (x + 2)]}$

Step 1.3.2

By the Sum Rule, the derivative of $x^{3} + 5 x^{2} + 6 x$ with respect to $x$ is $\frac{d}{d x} [x^{3}] + \frac{d}{d x} [5 x^{2}] + \frac{d}{d x} [6 x]$ .

$lim_{x \to - 2} \frac{\frac{d}{d x} [x^{3}] + \frac{d}{d x} [5 x^{2}] + \frac{d}{d x} [6 x]}{\frac{d}{d x} [x^{2} (x + 2) - (x + 2)]}$

Step 1.3.3

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

$lim_{x \to - 2} \frac{3 x^{2} + \frac{d}{d x} [5 x^{2}] + \frac{d}{d x} [6 x]}{\frac{d}{d x} [x^{2} (x + 2) - (x + 2)]}$

Step 1.3.4

Evaluate $\frac{d}{d x} [5 x^{2}]$ .

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Step 1.3.4.1

Since $5$ is constant with respect to $x$ , the derivative of $5 x^{2}$ with respect to $x$ is $5 \frac{d}{d x} [x^{2}]$ .

$lim_{x \to - 2} \frac{3 x^{2} + 5 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [6 x]}{\frac{d}{d x} [x^{2} (x + 2) - (x + 2)]}$

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 $n = 2$ .

$lim_{x \to - 2} \frac{3 x^{2} + 5 (2 x) + \frac{d}{d x} [6 x]}{\frac{d}{d x} [x^{2} (x + 2) - (x + 2)]}$

Step 1.3.4.3

Multiply $2$ by $5$ .

$lim_{x \to - 2} \frac{3 x^{2} + 10 x + \frac{d}{d x} [6 x]}{\frac{d}{d x} [x^{2} (x + 2) - (x + 2)]}$

Step 1.3.5

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

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Step 1.3.5.1

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

$lim_{x \to - 2} \frac{3 x^{2} + 10 x + 6 \frac{d}{d x} [x]}{\frac{d}{d x} [x^{2} (x + 2) - (x + 2)]}$

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 $n = 1$ .

$lim_{x \to - 2} \frac{3 x^{2} + 10 x + 6 \cdot 1}{\frac{d}{d x} [x^{2} (x + 2) - (x + 2)]}$

Step 1.3.5.3

Multiply $6$ by $1$ .

$lim_{x \to - 2} \frac{3 x^{2} + 10 x + 6}{\frac{d}{d x} [x^{2} (x + 2) - (x + 2)]}$

Step 1.3.6

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

$lim_{x \to - 2} \frac{3 x^{2} + 10 x + 6}{\frac{d}{d x} [x^{2} (x + 2)] + \frac{d}{d x} [- (x + 2)]}$

Step 1.3.7

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

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Step 1.3.7.1

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 $g (x) = x + 2$ .

$lim_{x \to - 2} \frac{3 x^{2} + 10 x + 6}{x^{2} \frac{d}{d x} [x + 2] + (x + 2) \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- (x + 2)]}$

Step 1.3.7.2

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

$lim_{x \to - 2} \frac{3 x^{2} + 10 x + 6}{x^{2} (\frac{d}{d x} [x] + \frac{d}{d x} [2]) + (x + 2) \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- (x + 2)]}$

Step 1.3.7.3

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

$lim_{x \to - 2} \frac{3 x^{2} + 10 x + 6}{x^{2} (1 + \frac{d}{d x} [2]) + (x + 2) \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- (x + 2)]}$

Step 1.3.7.4

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

$lim_{x \to - 2} \frac{3 x^{2} + 10 x + 6}{x^{2} (1 + 0) + (x + 2) \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- (x + 2)]}$

Step 1.3.7.5

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

$lim_{x \to - 2} \frac{3 x^{2} + 10 x + 6}{x^{2} (1 + 0) + (x + 2) (2 x) + \frac{d}{d x} [- (x + 2)]}$

Step 1.3.7.6

Add $1$ and $0$ .

$lim_{x \to - 2} \frac{3 x^{2} + 10 x + 6}{x^{2} \cdot 1 + (x + 2) (2 x) + \frac{d}{d x} [- (x + 2)]}$

Step 1.3.7.7

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

$lim_{x \to - 2} \frac{3 x^{2} + 10 x + 6}{x^{2} + (x + 2) (2 x) + \frac{d}{d x} [- (x + 2)]}$

Step 1.3.7.8

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

$lim_{x \to - 2} \frac{3 x^{2} + 10 x + 6}{x^{2} + 2 (x + 2) x + \frac{d}{d x} [- (x + 2)]}$

Step 1.3.8

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

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Step 1.3.8.1

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

$lim_{x \to - 2} \frac{3 x^{2} + 10 x + 6}{x^{2} + 2 (x + 2) x - \frac{d}{d x} [x + 2]}$

Step 1.3.8.2

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

$lim_{x \to - 2} \frac{3 x^{2} + 10 x + 6}{x^{2} + 2 (x + 2) x - (\frac{d}{d x} [x] + \frac{d}{d x} [2])}$

Step 1.3.8.3

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

$lim_{x \to - 2} \frac{3 x^{2} + 10 x + 6}{x^{2} + 2 (x + 2) x - (1 + \frac{d}{d x} [2])}$

Step 1.3.8.4

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

$lim_{x \to - 2} \frac{3 x^{2} + 10 x + 6}{x^{2} + 2 (x + 2) x - (1 + 0)}$

Step 1.3.8.5

Add $1$ and $0$ .

$lim_{x \to - 2} \frac{3 x^{2} + 10 x + 6}{x^{2} + 2 (x + 2) x - 1 \cdot 1}$

Step 1.3.8.6

Multiply $- 1$ by $1$ .

$lim_{x \to - 2} \frac{3 x^{2} + 10 x + 6}{x^{2} + 2 (x + 2) x - 1}$

Step 1.3.9

Simplify.

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Step 1.3.9.1

Apply the distributive property.

$lim_{x \to - 2} \frac{3 x^{2} + 10 x + 6}{x^{2} + (2 x + 2 \cdot 2) x - 1}$

Step 1.3.9.2

Apply the distributive property.

$lim_{x \to - 2} \frac{3 x^{2} + 10 x + 6}{x^{2} + 2 x \cdot x + 2 \cdot 2 x - 1}$

Step 1.3.9.3

Combine terms.

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Step 1.3.9.3.1

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

$lim_{x \to - 2} \frac{3 x^{2} + 10 x + 6}{x^{2} + 2 (x^{1} x) + 2 \cdot 2 x - 1}$

Step 1.3.9.3.2

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

$lim_{x \to - 2} \frac{3 x^{2} + 10 x + 6}{x^{2} + 2 (x^{1} x^{1}) + 2 \cdot 2 x - 1}$

Step 1.3.9.3.3

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

$lim_{x \to - 2} \frac{3 x^{2} + 10 x + 6}{x^{2} + 2 x^{1 + 1} + 2 \cdot 2 x - 1}$

Step 1.3.9.3.4

Add $1$ and $1$ .

$lim_{x \to - 2} \frac{3 x^{2} + 10 x + 6}{x^{2} + 2 x^{2} + 2 \cdot 2 x - 1}$

Step 1.3.9.3.5

Multiply $2$ by $2$ .

$lim_{x \to - 2} \frac{3 x^{2} + 10 x + 6}{x^{2} + 2 x^{2} + 4 x - 1}$

Step 1.3.9.3.6

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

$lim_{x \to - 2} \frac{3 x^{2} + 10 x + 6}{3 x^{2} + 4 x - 1}$

Step 2

Evaluate the limit.

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Step 2.1

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

$\frac{lim_{x \to - 2} 3 x^{2} + 10 x + 6}{lim_{x \to - 2} 3 x^{2} + 4 x - 1}$

Step 2.2

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

$\frac{lim_{x \to - 2} 3 x^{2} + lim_{x \to - 2} 10 x + lim_{x \to - 2} 6}{lim_{x \to - 2} 3 x^{2} + 4 x - 1}$

Step 2.3

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

$\frac{3 lim_{x \to - 2} x^{2} + lim_{x \to - 2} 10 x + lim_{x \to - 2} 6}{lim_{x \to - 2} 3 x^{2} + 4 x - 1}$

Step 2.4

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

$\frac{3 {(lim_{x \to - 2} x)}^{2} + lim_{x \to - 2} 10 x + lim_{x \to - 2} 6}{lim_{x \to - 2} 3 x^{2} + 4 x - 1}$

Step 2.5

Move the term $10$ outside of the limit because it is constant with respect to $x$ .

$\frac{3 {(lim_{x \to - 2} x)}^{2} + 10 lim_{x \to - 2} x + lim_{x \to - 2} 6}{lim_{x \to - 2} 3 x^{2} + 4 x - 1}$

Step 2.6

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

$\frac{3 {(lim_{x \to - 2} x)}^{2} + 10 lim_{x \to - 2} x + 6}{lim_{x \to - 2} 3 x^{2} + 4 x - 1}$

Step 2.7

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

$\frac{3 {(lim_{x \to - 2} x)}^{2} + 10 lim_{x \to - 2} x + 6}{lim_{x \to - 2} 3 x^{2} + lim_{x \to - 2} 4 x - lim_{x \to - 2} 1}$

Step 2.8

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

$\frac{3 {(lim_{x \to - 2} x)}^{2} + 10 lim_{x \to - 2} x + 6}{3 lim_{x \to - 2} x^{2} + lim_{x \to - 2} 4 x - lim_{x \to - 2} 1}$

Step 2.9

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

$\frac{3 {(lim_{x \to - 2} x)}^{2} + 10 lim_{x \to - 2} x + 6}{3 {(lim_{x \to - 2} x)}^{2} + lim_{x \to - 2} 4 x - lim_{x \to - 2} 1}$

Step 2.10

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

$\frac{3 {(lim_{x \to - 2} x)}^{2} + 10 lim_{x \to - 2} x + 6}{3 {(lim_{x \to - 2} x)}^{2} + 4 lim_{x \to - 2} x - lim_{x \to - 2} 1}$

Step 2.11

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

$\frac{3 {(lim_{x \to - 2} x)}^{2} + 10 lim_{x \to - 2} x + 6}{3 {(lim_{x \to - 2} x)}^{2} + 4 lim_{x \to - 2} x - 1 \cdot 1}$

Step 3

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

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Step 3.1

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

$\frac{3 {(- 2)}^{2} + 10 lim_{x \to - 2} x + 6}{3 {(lim_{x \to - 2} x)}^{2} + 4 lim_{x \to - 2} x - 1 \cdot 1}$

Step 3.2

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

$\frac{3 {(- 2)}^{2} + 10 \cdot - 2 + 6}{3 {(lim_{x \to - 2} x)}^{2} + 4 lim_{x \to - 2} x - 1 \cdot 1}$

Step 3.3

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

$\frac{3 {(- 2)}^{2} + 10 \cdot - 2 + 6}{3 {(- 2)}^{2} + 4 lim_{x \to - 2} x - 1 \cdot 1}$

Step 3.4

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

$\frac{3 {(- 2)}^{2} + 10 \cdot - 2 + 6}{3 {(- 2)}^{2} + 4 \cdot - 2 - 1 \cdot 1}$

Step 4

Simplify the answer.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

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

$\frac{3 \cdot 4 + 10 \cdot - 2 + 6}{3 {(- 2)}^{2} + 4 \cdot - 2 - 1 \cdot 1}$

Step 4.1.2

Multiply $3$ by $4$ .

$\frac{12 + 10 \cdot - 2 + 6}{3 {(- 2)}^{2} + 4 \cdot - 2 - 1 \cdot 1}$

Step 4.1.3

Multiply $10$ by $- 2$ .

$\frac{12 - 20 + 6}{3 {(- 2)}^{2} + 4 \cdot - 2 - 1 \cdot 1}$

Step 4.1.4

Subtract $20$ from $12$ .

$\frac{- 8 + 6}{3 {(- 2)}^{2} + 4 \cdot - 2 - 1 \cdot 1}$

Step 4.1.5

Add $- 8$ and $6$ .

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

Step 4.2

Simplify the denominator.

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Step 4.2.1

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

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

Step 4.2.2

Multiply $3$ by $4$ .

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

Step 4.2.3

Multiply $4$ by $- 2$ .

$\frac{- 2}{12 - 8 - 1 \cdot 1}$

Step 4.2.4

Multiply $- 1$ by $1$ .

$\frac{- 2}{12 - 8 - 1}$

Step 4.2.5

Subtract $8$ from $12$ .

$\frac{- 2}{4 - 1}$

Step 4.2.6

Subtract $1$ from $4$ .

$\frac{- 2}{3}$

Step 4.3

Move the negative in front of the fraction.

$- \frac{2}{3}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$- \frac{2}{3}$

Decimal Form:

$- 0.\overline{6}$