Calculus Examples

$lim_{x \to 3} \frac{x^{3} - 4 x - 15}{x^{3} + x^{2} - 6 x - 18}$

Step 1

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

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Step 1.1

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

$\frac{lim_{x \to 3} x^{3} - 4 x - 15}{lim_{x \to 3} x^{3} + x^{2} - 6 x - 18}$

Step 1.2

Evaluate the limit of the numerator.

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Step 1.2.1

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

$\frac{lim_{x \to 3} x^{3} - lim_{x \to 3} 4 x - lim_{x \to 3} 15}{lim_{x \to 3} x^{3} + x^{2} - 6 x - 18}$

Step 1.2.2

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

$\frac{{(lim_{x \to 3} x)}^{3} - lim_{x \to 3} 4 x - lim_{x \to 3} 15}{lim_{x \to 3} x^{3} + x^{2} - 6 x - 18}$

Step 1.2.3

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

$\frac{{(lim_{x \to 3} x)}^{3} - 4 lim_{x \to 3} x - lim_{x \to 3} 15}{lim_{x \to 3} x^{3} + x^{2} - 6 x - 18}$

Step 1.2.4

Evaluate the limit of $15$ which is constant as $x$ approaches $3$ .

$\frac{{(lim_{x \to 3} x)}^{3} - 4 lim_{x \to 3} x - 1 \cdot 15}{lim_{x \to 3} x^{3} + x^{2} - 6 x - 18}$

Step 1.2.5

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

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Step 1.2.5.1

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

$\frac{3^{3} - 4 lim_{x \to 3} x - 1 \cdot 15}{lim_{x \to 3} x^{3} + x^{2} - 6 x - 18}$

Step 1.2.5.2

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

$\frac{3^{3} - 4 \cdot 3 - 1 \cdot 15}{lim_{x \to 3} x^{3} + x^{2} - 6 x - 18}$

Step 1.2.6

Simplify the answer.

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Step 1.2.6.1

Simplify each term.

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Step 1.2.6.1.1

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

$\frac{27 - 4 \cdot 3 - 1 \cdot 15}{lim_{x \to 3} x^{3} + x^{2} - 6 x - 18}$

Step 1.2.6.1.2

Multiply $- 4$ by $3$ .

$\frac{27 - 12 - 1 \cdot 15}{lim_{x \to 3} x^{3} + x^{2} - 6 x - 18}$

Step 1.2.6.1.3

Multiply $- 1$ by $15$ .

$\frac{27 - 12 - 15}{lim_{x \to 3} x^{3} + x^{2} - 6 x - 18}$

Step 1.2.6.2

Subtract $12$ from $27$ .

$\frac{15 - 15}{lim_{x \to 3} x^{3} + x^{2} - 6 x - 18}$

Step 1.2.6.3

Subtract $15$ from $15$ .

$\frac{0}{lim_{x \to 3} x^{3} + x^{2} - 6 x - 18}$

Step 1.3

Evaluate the limit of the denominator.

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Step 1.3.1

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

$\frac{0}{lim_{x \to 3} x^{3} + lim_{x \to 3} x^{2} - lim_{x \to 3} 6 x - lim_{x \to 3} 18}$

Step 1.3.2

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

$\frac{0}{{(lim_{x \to 3} x)}^{3} + lim_{x \to 3} x^{2} - lim_{x \to 3} 6 x - lim_{x \to 3} 18}$

Step 1.3.3

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

$\frac{0}{{(lim_{x \to 3} x)}^{3} + {(lim_{x \to 3} x)}^{2} - lim_{x \to 3} 6 x - lim_{x \to 3} 18}$

Step 1.3.4

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

$\frac{0}{{(lim_{x \to 3} x)}^{3} + {(lim_{x \to 3} x)}^{2} - 6 lim_{x \to 3} x - lim_{x \to 3} 18}$

Step 1.3.5

Evaluate the limit of $18$ which is constant as $x$ approaches $3$ .

$\frac{0}{{(lim_{x \to 3} x)}^{3} + {(lim_{x \to 3} x)}^{2} - 6 lim_{x \to 3} x - 1 \cdot 18}$

Step 1.3.6

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

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Step 1.3.6.1

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

$\frac{0}{3^{3} + {(lim_{x \to 3} x)}^{2} - 6 lim_{x \to 3} x - 1 \cdot 18}$

Step 1.3.6.2

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

$\frac{0}{3^{3} + 3^{2} - 6 lim_{x \to 3} x - 1 \cdot 18}$

Step 1.3.6.3

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

$\frac{0}{3^{3} + 3^{2} - 6 \cdot 3 - 1 \cdot 18}$

Step 1.3.7

Simplify the answer.

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Step 1.3.7.1

Simplify each term.

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Step 1.3.7.1.1

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

$\frac{0}{27 + 3^{2} - 6 \cdot 3 - 1 \cdot 18}$

Step 1.3.7.1.2

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

$\frac{0}{27 + 9 - 6 \cdot 3 - 1 \cdot 18}$

Step 1.3.7.1.3

Multiply $- 6$ by $3$ .

$\frac{0}{27 + 9 - 18 - 1 \cdot 18}$

Step 1.3.7.1.4

Multiply $- 1$ by $18$ .

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

Step 1.3.7.2

Add $27$ and $9$ .

$\frac{0}{36 - 18 - 18}$

Step 1.3.7.3

Subtract $18$ from $36$ .

$\frac{0}{18 - 18}$

Step 1.3.7.4

Subtract $18$ from $18$ .

$\frac{0}{0}$

Step 1.3.7.5

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

Undefined

$\frac{0}{0}$

Step 1.3.8

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

Undefined

$\frac{0}{0}$

Step 1.4

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

Undefined

$\frac{0}{0}$

Step 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.

$lim_{x \to 3} \frac{x^{3} - 4 x - 15}{x^{3} + x^{2} - 6 x - 18} = lim_{x \to 3} \frac{\frac{d}{d x} [x^{3} - 4 x - 15]}{\frac{d}{d x} [x^{3} + x^{2} - 6 x - 18]}$

Step 3

Find the derivative of the numerator and denominator.

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Step 3.1

Differentiate the numerator and denominator.

$lim_{x \to 3} \frac{\frac{d}{d x} [x^{3} - 4 x - 15]}{\frac{d}{d x} [x^{3} + x^{2} - 6 x - 18]}$

Step 3.2

By the Sum Rule, the derivative of $x^{3} - 4 x - 15$ with respect to $x$ is $\frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 4 x] + \frac{d}{d x} [- 15]$ .

$lim_{x \to 3} \frac{\frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 4 x] + \frac{d}{d x} [- 15]}{\frac{d}{d x} [x^{3} + x^{2} - 6 x - 18]}$

Step 3.3

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

$lim_{x \to 3} \frac{3 x^{2} + \frac{d}{d x} [- 4 x] + \frac{d}{d x} [- 15]}{\frac{d}{d x} [x^{3} + x^{2} - 6 x - 18]}$

Step 3.4

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

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Step 3.4.1

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

$lim_{x \to 3} \frac{3 x^{2} - 4 \frac{d}{d x} [x] + \frac{d}{d x} [- 15]}{\frac{d}{d x} [x^{3} + x^{2} - 6 x - 18]}$

Step 3.4.2

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

$lim_{x \to 3} \frac{3 x^{2} - 4 \cdot 1 + \frac{d}{d x} [- 15]}{\frac{d}{d x} [x^{3} + x^{2} - 6 x - 18]}$

Step 3.4.3

Multiply $- 4$ by $1$ .

$lim_{x \to 3} \frac{3 x^{2} - 4 + \frac{d}{d x} [- 15]}{\frac{d}{d x} [x^{3} + x^{2} - 6 x - 18]}$

Step 3.5

Since $- 15$ is constant with respect to $x$ , the derivative of $- 15$ with respect to $x$ is $0$ .

$lim_{x \to 3} \frac{3 x^{2} - 4 + 0}{\frac{d}{d x} [x^{3} + x^{2} - 6 x - 18]}$

Step 3.6

Add $3 x^{2} - 4$ and $0$ .

$lim_{x \to 3} \frac{3 x^{2} - 4}{\frac{d}{d x} [x^{3} + x^{2} - 6 x - 18]}$

Step 3.7

By the Sum Rule, the derivative of $x^{3} + x^{2} - 6 x - 18$ with respect to $x$ is $\frac{d}{d x} [x^{3}] + \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 6 x] + \frac{d}{d x} [- 18]$ .

$lim_{x \to 3} \frac{3 x^{2} - 4}{\frac{d}{d x} [x^{3}] + \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 6 x] + \frac{d}{d x} [- 18]}$

Step 3.8

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

$lim_{x \to 3} \frac{3 x^{2} - 4}{3 x^{2} + \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 6 x] + \frac{d}{d x} [- 18]}$

Step 3.9

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

$lim_{x \to 3} \frac{3 x^{2} - 4}{3 x^{2} + 2 x + \frac{d}{d x} [- 6 x] + \frac{d}{d x} [- 18]}$

Step 3.10

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

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Step 3.10.1

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

$lim_{x \to 3} \frac{3 x^{2} - 4}{3 x^{2} + 2 x - 6 \frac{d}{d x} [x] + \frac{d}{d x} [- 18]}$

Step 3.10.2

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

$lim_{x \to 3} \frac{3 x^{2} - 4}{3 x^{2} + 2 x - 6 \cdot 1 + \frac{d}{d x} [- 18]}$

Step 3.10.3

Multiply $- 6$ by $1$ .

$lim_{x \to 3} \frac{3 x^{2} - 4}{3 x^{2} + 2 x - 6 + \frac{d}{d x} [- 18]}$

Step 3.11

Since $- 18$ is constant with respect to $x$ , the derivative of $- 18$ with respect to $x$ is $0$ .

$lim_{x \to 3} \frac{3 x^{2} - 4}{3 x^{2} + 2 x - 6 + 0}$

Step 3.12

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

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

Step 4

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $3$ .

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

Step 5

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

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

Step 6

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

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

Step 7

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

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

Step 8

Evaluate the limit of $4$ which is constant as $x$ approaches $3$ .

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

Step 9

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

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

Step 10

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

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

Step 11

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

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

Step 12

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

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

Step 13

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

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

Step 14

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

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Step 14.1

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

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

Step 14.2

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

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

Step 14.3

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

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

Step 15

Simplify the answer.

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Step 15.1

Simplify the numerator.

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Step 15.1.1

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

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Step 15.1.1.1

Multiply $3$ by $3^{2}$ .

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Step 15.1.1.1.1

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

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

Step 15.1.1.1.2

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

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

Step 15.1.1.2

Add $1$ and $2$ .

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

Step 15.1.2

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

$\frac{27 - 1 \cdot 4}{3 \cdot 3^{2} + 2 \cdot 3 - 1 \cdot 6}$

Step 15.1.3

Multiply $- 1$ by $4$ .

$\frac{27 - 4}{3 \cdot 3^{2} + 2 \cdot 3 - 1 \cdot 6}$

Step 15.1.4

Subtract $4$ from $27$ .

$\frac{23}{3 \cdot 3^{2} + 2 \cdot 3 - 1 \cdot 6}$

Step 15.2

Simplify the denominator.

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Step 15.2.1

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

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Step 15.2.1.1

Multiply $3$ by $3^{2}$ .

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Step 15.2.1.1.1

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

$\frac{23}{3^{1} \cdot 3^{2} + 2 \cdot 3 - 1 \cdot 6}$

Step 15.2.1.1.2

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

$\frac{23}{3^{1 + 2} + 2 \cdot 3 - 1 \cdot 6}$

Step 15.2.1.2

Add $1$ and $2$ .

$\frac{23}{3^{3} + 2 \cdot 3 - 1 \cdot 6}$

Step 15.2.2

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

$\frac{23}{27 + 2 \cdot 3 - 1 \cdot 6}$

Step 15.2.3

Multiply $2$ by $3$ .

$\frac{23}{27 + 6 - 1 \cdot 6}$

Step 15.2.4

Multiply $- 1$ by $6$ .

$\frac{23}{27 + 6 - 6}$

Step 15.2.5

Add $27$ and $6$ .

$\frac{23}{33 - 6}$

Step 15.2.6

Subtract $6$ from $33$ .

$\frac{23}{27}$

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