Calculus Examples

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

Step 1

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $6$ .

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

Step 2

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

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

Step 3

Simplify terms.

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Step 3.1

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

$\frac{6 - 6}{| 36 - x^{2} |}$

Step 3.2

Simplify the answer.

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Step 3.2.1

Subtract $6$ from $6$ .

$\frac{0}{| 36 - x^{2} |}$

Step 3.2.2

Simplify the denominator.

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Step 3.2.2.1

Rewrite $36$ as $6^{2}$ .

$\frac{0}{| 6^{2} - x^{2} |}$

Step 3.2.2.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 6$ and $b = x$ .

$\frac{0}{| (6 + x) (6 - x) |}$

Step 3.2.2.3

Expand $(6 + x) (6 - x)$ using the FOIL Method.

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Step 3.2.2.3.1

Apply the distributive property.

$\frac{0}{| 6 (6 - x) + x (6 - x) |}$

Step 3.2.2.3.2

Apply the distributive property.

$\frac{0}{| 6 \cdot 6 + 6 (- x) + x (6 - x) |}$

Step 3.2.2.3.3

Apply the distributive property.

$\frac{0}{| 6 \cdot 6 + 6 (- x) + x \cdot 6 + x (- x) |}$

Step 3.2.2.4

Simplify and combine like terms.

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Step 3.2.2.4.1

Simplify each term.

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Step 3.2.2.4.1.1

Multiply $6$ by $6$ .

$\frac{0}{| 36 + 6 (- x) + x \cdot 6 + x (- x) |}$

Step 3.2.2.4.1.2

Multiply $- 1$ by $6$ .

$\frac{0}{| 36 - 6 x + x \cdot 6 + x (- x) |}$

Step 3.2.2.4.1.3

Move $6$ to the left of $x$ .

$\frac{0}{| 36 - 6 x + 6 \cdot x + x (- x) |}$

Step 3.2.2.4.1.4

Rewrite using the commutative property of multiplication.

$\frac{0}{| 36 - 6 x + 6 x - x \cdot x |}$

Step 3.2.2.4.1.5

Multiply $x$ by $x$ by adding the exponents.

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Step 3.2.2.4.1.5.1

Move $x$ .

$\frac{0}{| 36 - 6 x + 6 x - (x \cdot x) |}$

Step 3.2.2.4.1.5.2

Multiply $x$ by $x$ .

$\frac{0}{| 36 - 6 x + 6 x - x^{2} |}$

Step 3.2.2.4.2

Add $- 6 x$ and $6 x$ .

$\frac{0}{| 36 + 0 - x^{2} |}$

Step 3.2.2.4.3

Add $36$ and $0$ .

$\frac{0}{| 36 - x^{2} |}$

Step 3.2.2.5

Rewrite $36$ as $6^{2}$ .

$\frac{0}{| 6^{2} - x^{2} |}$

Step 3.2.2.6

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 6$ and $b = x$ .

$\frac{0}{| (6 + x) (6 - x) |}$

Step 3.2.3

Divide $0$ by $| (6 + x) (6 - x) |$ .

$0$