Calculus Examples

$lim_{x \to 6} \frac{x^{3} - 36 x}{x - 6}$

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.

$\frac{lim_{x \to 6} x^{3} - 36 x}{lim_{x \to 6} x - 6}$

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 $6$ .

$\frac{lim_{x \to 6} x^{3} - lim_{x \to 6} 36 x}{lim_{x \to 6} x - 6}$

Step 1.1.2.2

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

$\frac{{(lim_{x \to 6} x)}^{3} - lim_{x \to 6} 36 x}{lim_{x \to 6} x - 6}$

Step 1.1.2.3

Move the term $36$ outside of the limit because it is constant with respect to $x$ .

$\frac{{(lim_{x \to 6} x)}^{3} - 36 lim_{x \to 6} x}{lim_{x \to 6} x - 6}$

Step 1.1.2.4

Evaluate the limits by plugging in $6$ for all occurrences of $x$ .

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Step 1.1.2.4.1

Evaluate the limit of $x$ by plugging in $6$ for $x$ .

$\frac{6^{3} - 36 lim_{x \to 6} x}{lim_{x \to 6} x - 6}$

Step 1.1.2.4.2

Evaluate the limit of $x$ by plugging in $6$ for $x$ .

$\frac{6^{3} - 36 \cdot 6}{lim_{x \to 6} x - 6}$

Step 1.1.2.5

Simplify the answer.

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Step 1.1.2.5.1

Simplify each term.

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Step 1.1.2.5.1.1

Raise $6$ to the power of $3$ .

$\frac{216 - 36 \cdot 6}{lim_{x \to 6} x - 6}$

Step 1.1.2.5.1.2

Multiply $- 36$ by $6$ .

$\frac{216 - 216}{lim_{x \to 6} x - 6}$

Step 1.1.2.5.2

Subtract $216$ from $216$ .

$\frac{0}{lim_{x \to 6} x - 6}$

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 $6$ .

$\frac{0}{lim_{x \to 6} x - lim_{x \to 6} 6}$

Step 1.1.3.1.2

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

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

Step 1.1.3.2

Evaluate the limit of $x$ by plugging in $6$ for $x$ .

$\frac{0}{6 - 1 \cdot 6}$

Step 1.1.3.3

Simplify the answer.

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Step 1.1.3.3.1

Multiply $- 1$ by $6$ .

$\frac{0}{6 - 6}$

Step 1.1.3.3.2

Subtract $6$ from $6$ .

$\frac{0}{0}$

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.

$lim_{x \to 6} \frac{x^{3} - 36 x}{x - 6} = lim_{x \to 6} \frac{\frac{d}{d x} [x^{3} - 36 x]}{\frac{d}{d x} [x - 6]}$

Step 1.3

Find the derivative of the numerator and denominator.

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Step 1.3.1

Differentiate the numerator and denominator.

$lim_{x \to 6} \frac{\frac{d}{d x} [x^{3} - 36 x]}{\frac{d}{d x} [x - 6]}$

Step 1.3.2

By the Sum Rule, the derivative of $x^{3} - 36 x$ with respect to $x$ is $\frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 36 x]$ .

$lim_{x \to 6} \frac{\frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 36 x]}{\frac{d}{d x} [x - 6]}$

Step 1.3.3

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

$lim_{x \to 6} \frac{3 x^{2} + \frac{d}{d x} [- 36 x]}{\frac{d}{d x} [x - 6]}$

Step 1.3.4

Evaluate $\frac{d}{d x} [- 36 x]$ .

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Step 1.3.4.1

Since $- 36$ is constant with respect to $x$ , the derivative of $- 36 x$ with respect to $x$ is $- 36 \frac{d}{d x} [x]$ .

$lim_{x \to 6} \frac{3 x^{2} - 36 \frac{d}{d x} [x]}{\frac{d}{d x} [x - 6]}$

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 $n = 1$ .

$lim_{x \to 6} \frac{3 x^{2} - 36 \cdot 1}{\frac{d}{d x} [x - 6]}$

Step 1.3.4.3

Multiply $- 36$ by $1$ .

$lim_{x \to 6} \frac{3 x^{2} - 36}{\frac{d}{d x} [x - 6]}$

Step 1.3.5

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

$lim_{x \to 6} \frac{3 x^{2} - 36}{\frac{d}{d x} [x] + \frac{d}{d x} [- 6]}$

Step 1.3.6

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

$lim_{x \to 6} \frac{3 x^{2} - 36}{1 + \frac{d}{d x} [- 6]}$

Step 1.3.7

Since $- 6$ is constant with respect to $x$ , the derivative of $- 6$ with respect to $x$ is $0$ .

$lim_{x \to 6} \frac{3 x^{2} - 36}{1 + 0}$

Step 1.3.8

Add $1$ and $0$ .

$lim_{x \to 6} \frac{3 x^{2} - 36}{1}$

Step 1.4

Divide $3 x^{2} - 36$ by $1$ .

$lim_{x \to 6} 3 x^{2} - 36$

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 $6$ .

$lim_{x \to 6} 3 x^{2} - lim_{x \to 6} 36$

Step 2.2

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

$3 lim_{x \to 6} x^{2} - lim_{x \to 6} 36$

Step 2.3

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

$3 {(lim_{x \to 6} x)}^{2} - lim_{x \to 6} 36$

Step 2.4

Evaluate the limit of $36$ which is constant as $x$ approaches $6$ .

$3 {(lim_{x \to 6} x)}^{2} - 1 \cdot 36$

Step 3

Evaluate the limit of $x$ by plugging in $6$ for $x$ .

$3 \cdot 6^{2} - 1 \cdot 36$

Step 4

Simplify the answer.

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Step 4.1

Simplify each term.

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Step 4.1.1

Raise $6$ to the power of $2$ .

$3 \cdot 36 - 1 \cdot 36$

Step 4.1.2

Multiply $3$ by $36$ .

$108 - 1 \cdot 36$

Step 4.1.3

Multiply $- 1$ by $36$ .

$108 - 36$

Step 4.2

Subtract $36$ from $108$ .

$72$