Calculus Examples

$lim_{x \to 3} \frac{x^{3} - 3 x^{2}}{5 x - 15}$

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.

$\frac{lim_{x \to 3} x^{3} - 3 x^{2}}{lim_{x \to 3} 5 x - 15}$

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 $3$ .

$\frac{lim_{x \to 3} x^{3} - lim_{x \to 3} 3 x^{2}}{lim_{x \to 3} 5 x - 15}$

Step 1.1.2.2

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

$\frac{{(lim_{x \to 3} x)}^{3} - lim_{x \to 3} 3 x^{2}}{lim_{x \to 3} 5 x - 15}$

Step 1.1.2.3

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

$\frac{{(lim_{x \to 3} x)}^{3} - 3 lim_{x \to 3} x^{2}}{lim_{x \to 3} 5 x - 15}$

Step 1.1.2.4

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

$\frac{{(lim_{x \to 3} x)}^{3} - 3 {(lim_{x \to 3} x)}^{2}}{lim_{x \to 3} 5 x - 15}$

Step 1.1.2.5

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

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Step 1.1.2.5.1

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

$\frac{3^{3} - 3 {(lim_{x \to 3} x)}^{2}}{lim_{x \to 3} 5 x - 15}$

Step 1.1.2.5.2

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

$\frac{3^{3} - 3 \cdot 3^{2}}{lim_{x \to 3} 5 x - 15}$

Step 1.1.2.6

Simplify the answer.

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Step 1.1.2.6.1

Simplify each term.

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Step 1.1.2.6.1.1

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

$\frac{27 - 3 \cdot 3^{2}}{lim_{x \to 3} 5 x - 15}$

Step 1.1.2.6.1.2

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

$\frac{27 - 3 \cdot 9}{lim_{x \to 3} 5 x - 15}$

Step 1.1.2.6.1.3

Multiply $- 3$ by $9$ .

$\frac{27 - 27}{lim_{x \to 3} 5 x - 15}$

Step 1.1.2.6.2

Subtract $27$ from $27$ .

$\frac{0}{lim_{x \to 3} 5 x - 15}$

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 $3$ .

$\frac{0}{lim_{x \to 3} 5 x - lim_{x \to 3} 15}$

Step 1.1.3.1.2

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

$\frac{0}{5 lim_{x \to 3} x - lim_{x \to 3} 15}$

Step 1.1.3.1.3

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

$\frac{0}{5 lim_{x \to 3} x - 1 \cdot 15}$

Step 1.1.3.2

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

$\frac{0}{5 \cdot 3 - 1 \cdot 15}$

Step 1.1.3.3

Simplify the answer.

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Step 1.1.3.3.1

Simplify each term.

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Step 1.1.3.3.1.1

Multiply $5$ by $3$ .

$\frac{0}{15 - 1 \cdot 15}$

Step 1.1.3.3.1.2

Multiply $- 1$ by $15$ .

$\frac{0}{15 - 15}$

Step 1.1.3.3.2

Subtract $15$ from $15$ .

$\frac{0}{0}$

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.

$lim_{x \to 3} \frac{x^{3} - 3 x^{2}}{5 x - 15} = lim_{x \to 3} \frac{\frac{d}{d x} [x^{3} - 3 x^{2}]}{\frac{d}{d x} [5 x - 15]}$

Step 1.3

Find the derivative of the numerator and denominator.

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Step 1.3.1

Differentiate the numerator and denominator.

$lim_{x \to 3} \frac{\frac{d}{d x} [x^{3} - 3 x^{2}]}{\frac{d}{d x} [5 x - 15]}$

Step 1.3.2

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

$lim_{x \to 3} \frac{\frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 3 x^{2}]}{\frac{d}{d x} [5 x - 15]}$

Step 1.3.3

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

$lim_{x \to 3} \frac{3 x^{2} + \frac{d}{d x} [- 3 x^{2}]}{\frac{d}{d x} [5 x - 15]}$

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

$lim_{x \to 3} \frac{3 x^{2} - 3 \frac{d}{d x} [x^{2}]}{\frac{d}{d x} [5 x - 15]}$

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 $n = 2$ .

$lim_{x \to 3} \frac{3 x^{2} - 3 (2 x)}{\frac{d}{d x} [5 x - 15]}$

Step 1.3.4.3

Multiply $2$ by $- 3$ .

$lim_{x \to 3} \frac{3 x^{2} - 6 x}{\frac{d}{d x} [5 x - 15]}$

Step 1.3.5

By the Sum Rule, the derivative of $5 x - 15$ with respect to $x$ is $\frac{d}{d x} [5 x] + \frac{d}{d x} [- 15]$ .

$lim_{x \to 3} \frac{3 x^{2} - 6 x}{\frac{d}{d x} [5 x] + \frac{d}{d x} [- 15]}$

Step 1.3.6

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

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Step 1.3.6.1

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

$lim_{x \to 3} \frac{3 x^{2} - 6 x}{5 \frac{d}{d x} [x] + \frac{d}{d x} [- 15]}$

Step 1.3.6.2

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

$lim_{x \to 3} \frac{3 x^{2} - 6 x}{5 \cdot 1 + \frac{d}{d x} [- 15]}$

Step 1.3.6.3

Multiply $5$ by $1$ .

$lim_{x \to 3} \frac{3 x^{2} - 6 x}{5 + \frac{d}{d x} [- 15]}$

Step 1.3.7

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

$lim_{x \to 3} \frac{3 x^{2} - 6 x}{5 + 0}$

Step 1.3.8

Add $5$ and $0$ .

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

Step 2

Evaluate the limit.

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Step 2.1

Move the term $\frac{1}{5}$ outside of the limit because it is constant with respect to $x$ .

$\frac{1}{5} lim_{x \to 3} 3 x^{2} - 6 x$

Step 2.2

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

$\frac{1}{5} (lim_{x \to 3} 3 x^{2} - lim_{x \to 3} 6 x)$

Step 2.3

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

$\frac{1}{5} (3 lim_{x \to 3} x^{2} - lim_{x \to 3} 6 x)$

Step 2.4

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

$\frac{1}{5} (3 {(lim_{x \to 3} x)}^{2} - lim_{x \to 3} 6 x)$

Step 2.5

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

$\frac{1}{5} (3 {(lim_{x \to 3} x)}^{2} - 6 lim_{x \to 3} x)$

Step 3

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

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Step 3.1

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

$\frac{1}{5} (3 \cdot 3^{2} - 6 lim_{x \to 3} x)$

Step 3.2

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

$\frac{1}{5} (3 \cdot 3^{2} - 6 \cdot 3)$

Step 4

Simplify the answer.

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Step 4.1

Simplify each term.

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Step 4.1.1

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

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Step 4.1.1.1

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

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Step 4.1.1.1.1

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

$\frac{1}{5} (3^{1} \cdot 3^{2} - 6 \cdot 3)$

Step 4.1.1.1.2

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

$\frac{1}{5} (3^{1 + 2} - 6 \cdot 3)$

Step 4.1.1.2

Add $1$ and $2$ .

$\frac{1}{5} (3^{3} - 6 \cdot 3)$

Step 4.1.2

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

$\frac{1}{5} (27 - 6 \cdot 3)$

Step 4.1.3

Multiply $- 6$ by $3$ .

$\frac{1}{5} (27 - 18)$

Step 4.2

Subtract $18$ from $27$ .

$\frac{1}{5} \cdot 9$

Step 4.3

Combine $\frac{1}{5}$ and $9$ .

$\frac{9}{5}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\frac{9}{5}$

Decimal Form:

$1.8$