Calculus Examples

$lim_{x \to - 8} 1 - \frac{| x - x^{2} |}{\sqrt{x^{4} + | x |}}$

Step 1

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $- 8$ .

$lim_{x \to - 8} 1 - lim_{x \to - 8} \frac{| x - x^{2} |}{\sqrt{x^{4} + | x |}}$

Step 2

Evaluate the limit of $1$ which is constant as $x$ approaches $- 8$ .

$1 - lim_{x \to - 8} \frac{| x - x^{2} |}{\sqrt{x^{4} + | x |}}$

Step 3

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $- 8$ .

$1 - \frac{lim_{x \to - 8} | x - x^{2} |}{lim_{x \to - 8} \sqrt{x^{4} + | x |}}$

Step 4

Move the limit inside the absolute value signs.

$1 - \frac{| lim_{x \to - 8} x - x^{2} |}{lim_{x \to - 8} \sqrt{x^{4} + | x |}}$

Step 5

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $- 8$ .

$1 - \frac{| lim_{x \to - 8} x - lim_{x \to - 8} x^{2} |}{lim_{x \to - 8} \sqrt{x^{4} + | x |}}$

Step 6

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

$1 - \frac{| lim_{x \to - 8} x - {(lim_{x \to - 8} x)}^{2} |}{lim_{x \to - 8} \sqrt{x^{4} + | x |}}$

Step 7

Move the limit under the radical sign.

$1 - \frac{| lim_{x \to - 8} x - {(lim_{x \to - 8} x)}^{2} |}{\sqrt{lim_{x \to - 8} x^{4} + | x |}}$

Step 8

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $- 8$ .

$1 - \frac{| lim_{x \to - 8} x - {(lim_{x \to - 8} x)}^{2} |}{\sqrt{lim_{x \to - 8} x^{4} + lim_{x \to - 8} | x |}}$

Step 9

Move the exponent $4$ from $x^{4}$ outside the limit using the Limits Power Rule.

$1 - \frac{| lim_{x \to - 8} x - {(lim_{x \to - 8} x)}^{2} |}{\sqrt{{(lim_{x \to - 8} x)}^{4} + lim_{x \to - 8} | x |}}$

Step 10

Move the limit inside the absolute value signs.

$1 - \frac{| lim_{x \to - 8} x - {(lim_{x \to - 8} x)}^{2} |}{\sqrt{{(lim_{x \to - 8} x)}^{4} + | lim_{x \to - 8} x |}}$

Step 11

Evaluate the limits by plugging in $- 8$ for all occurrences of $x$ .

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Step 11.1

Evaluate the limit of $x$ by plugging in $- 8$ for $x$ .

$1 - \frac{| - 8 - {(lim_{x \to - 8} x)}^{2} |}{\sqrt{{(lim_{x \to - 8} x)}^{4} + | lim_{x \to - 8} x |}}$

Step 11.2

Evaluate the limit of $x$ by plugging in $- 8$ for $x$ .

$1 - \frac{| - 8 - {(- 8)}^{2} |}{\sqrt{{(lim_{x \to - 8} x)}^{4} + | lim_{x \to - 8} x |}}$

Step 11.3

Evaluate the limit of $x$ by plugging in $- 8$ for $x$ .

$1 - \frac{| - 8 - {(- 8)}^{2} |}{\sqrt{{(- 8)}^{4} + | lim_{x \to - 8} x |}}$

Step 11.4

Evaluate the limit of $x$ by plugging in $- 8$ for $x$ .

$1 - \frac{| - 8 - {(- 8)}^{2} |}{\sqrt{{(- 8)}^{4} + | - 8 |}}$

Step 12

Simplify each term.

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Step 12.1

Simplify the numerator.

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Step 12.1.1

Raise $- 8$ to the power of $2$ .

$1 - \frac{| - 8 - 1 \cdot 64 |}{\sqrt{{(- 8)}^{4} + | - 8 |}}$

Step 12.1.2

Multiply $- 1$ by $64$ .

$1 - \frac{| - 8 - 64 |}{\sqrt{{(- 8)}^{4} + | - 8 |}}$

Step 12.1.3

Subtract $64$ from $- 8$ .

$1 - \frac{| - 72 |}{\sqrt{{(- 8)}^{4} + | - 8 |}}$

Step 12.1.4

The absolute value is the distance between a number and zero. The distance between $- 72$ and $0$ is $72$ .

$1 - \frac{72}{\sqrt{{(- 8)}^{4} + | - 8 |}}$

Step 12.2

Simplify the denominator.

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Step 12.2.1

Raise $- 8$ to the power of $4$ .

$1 - \frac{72}{\sqrt{4096 + | - 8 |}}$

Step 12.2.2

The absolute value is the distance between a number and zero. The distance between $- 8$ and $0$ is $8$ .

$1 - \frac{72}{\sqrt{4096 + 8}}$

Step 12.2.3

Add $4096$ and $8$ .

$1 - \frac{72}{\sqrt{4104}}$

Step 12.2.4

Rewrite $4104$ as $6^{2} \cdot 114$ .

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Step 12.2.4.1

Factor $36$ out of $4104$ .

$1 - \frac{72}{\sqrt{36 (114)}}$

Step 12.2.4.2

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

$1 - \frac{72}{\sqrt{6^{2} \cdot 114}}$

Step 12.2.5

Pull terms out from under the radical.

$1 - \frac{72}{6 \sqrt{114}}$

Step 12.3

Cancel the common factor of $72$ and $6$ .

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Step 12.3.1

Factor $6$ out of $72$ .

$1 - \frac{6 \cdot 12}{6 \sqrt{114}}$

Step 12.3.2

Cancel the common factors.

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Step 12.3.2.1

Factor $6$ out of $6 \sqrt{114}$ .

$1 - \frac{6 \cdot 12}{6 (\sqrt{114})}$

Step 12.3.2.2

Cancel the common factor.

$1 - \frac{6 \cdot 12}{6 \sqrt{114}}$

Step 12.3.2.3

Rewrite the expression.

$1 - \frac{12}{\sqrt{114}}$

Step 12.4

Multiply $\frac{12}{\sqrt{114}}$ by $\frac{\sqrt{114}}{\sqrt{114}}$ .

$1 - (\frac{12}{\sqrt{114}} \cdot \frac{\sqrt{114}}{\sqrt{114}})$

Step 12.5

Combine and simplify the denominator.

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Step 12.5.1

Multiply $\frac{12}{\sqrt{114}}$ by $\frac{\sqrt{114}}{\sqrt{114}}$ .

$1 - \frac{12 \sqrt{114}}{\sqrt{114} \sqrt{114}}$

Step 12.5.2

Raise $\sqrt{114}$ to the power of $1$ .

$1 - \frac{12 \sqrt{114}}{{\sqrt{114}}^{1} \sqrt{114}}$

Step 12.5.3

Raise $\sqrt{114}$ to the power of $1$ .

$1 - \frac{12 \sqrt{114}}{{\sqrt{114}}^{1} {\sqrt{114}}^{1}}$

Step 12.5.4

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

$1 - \frac{12 \sqrt{114}}{{\sqrt{114}}^{1 + 1}}$

Step 12.5.5

Add $1$ and $1$ .

$1 - \frac{12 \sqrt{114}}{{\sqrt{114}}^{2}}$

Step 12.5.6

Rewrite ${\sqrt{114}}^{2}$ as $114$ .

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Step 12.5.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{114}$ as $114^{\frac{1}{2}}$ .

$1 - \frac{12 \sqrt{114}}{{(114^{\frac{1}{2}})}^{2}}$

Step 12.5.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$1 - \frac{12 \sqrt{114}}{114^{\frac{1}{2} \cdot 2}}$

Step 12.5.6.3

Combine $\frac{1}{2}$ and $2$ .

$1 - \frac{12 \sqrt{114}}{114^{\frac{2}{2}}}$

Step 12.5.6.4

Cancel the common factor of $2$ .

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Step 12.5.6.4.1

Cancel the common factor.

$1 - \frac{12 \sqrt{114}}{114^{\frac{2}{2}}}$

Step 12.5.6.4.2

Rewrite the expression.

$1 - \frac{12 \sqrt{114}}{114^{1}}$

Step 12.5.6.5

Evaluate the exponent.

$1 - \frac{12 \sqrt{114}}{114}$

Step 12.6

Cancel the common factor of $12$ and $114$ .

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Step 12.6.1

Factor $6$ out of $12 \sqrt{114}$ .

$1 - \frac{6 (2 \sqrt{114})}{114}$

Step 12.6.2

Cancel the common factors.

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Step 12.6.2.1

Factor $6$ out of $114$ .

$1 - \frac{6 (2 \sqrt{114})}{6 (19)}$

Step 12.6.2.2

Cancel the common factor.

$1 - \frac{6 (2 \sqrt{114})}{6 \cdot 19}$

Step 12.6.2.3

Rewrite the expression.

$1 - \frac{2 \sqrt{114}}{19}$

Step 13

The result can be shown in multiple forms.

Exact Form:

$1 - \frac{2 \sqrt{114}}{19}$

Decimal Form:

$- 0.12390297 \dots$