Calculus Examples

$lim_{x \to - 8} \frac{- 6 x}{x^{2} - \sqrt{x^{2} - 6 x}}$

Step 1

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

$- 6 lim_{x \to - 8} \frac{x}{x^{2} - \sqrt{x^{2} - 6 x}}$

Step 2

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

$- 6 \frac{lim_{x \to - 8} x}{lim_{x \to - 8} x^{2} - \sqrt{x^{2} - 6 x}}$

Step 3

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

$- 6 \frac{lim_{x \to - 8} x}{lim_{x \to - 8} x^{2} - lim_{x \to - 8} \sqrt{x^{2} - 6 x}}$

Step 4

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

$- 6 \frac{lim_{x \to - 8} x}{{(lim_{x \to - 8} x)}^{2} - lim_{x \to - 8} \sqrt{x^{2} - 6 x}}$

Step 5

Move the limit under the radical sign.

$- 6 \frac{lim_{x \to - 8} x}{{(lim_{x \to - 8} x)}^{2} - \sqrt{lim_{x \to - 8} x^{2} - 6 x}}$

Step 6

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

$- 6 \frac{lim_{x \to - 8} x}{{(lim_{x \to - 8} x)}^{2} - \sqrt{lim_{x \to - 8} x^{2} - lim_{x \to - 8} 6 x}}$

Step 7

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

$- 6 \frac{lim_{x \to - 8} x}{{(lim_{x \to - 8} x)}^{2} - \sqrt{{(lim_{x \to - 8} x)}^{2} - lim_{x \to - 8} 6 x}}$

Step 8

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

$- 6 \frac{lim_{x \to - 8} x}{{(lim_{x \to - 8} x)}^{2} - \sqrt{{(lim_{x \to - 8} x)}^{2} - 6 lim_{x \to - 8} x}}$

Step 9

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

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Step 9.1

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

$- 6 \frac{- 8}{{(lim_{x \to - 8} x)}^{2} - \sqrt{{(lim_{x \to - 8} x)}^{2} - 6 lim_{x \to - 8} x}}$

Step 9.2

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

$- 6 \frac{- 8}{{(- 8)}^{2} - \sqrt{{(lim_{x \to - 8} x)}^{2} - 6 lim_{x \to - 8} x}}$

Step 9.3

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

$- 6 \frac{- 8}{{(- 8)}^{2} - \sqrt{{(- 8)}^{2} - 6 lim_{x \to - 8} x}}$

Step 9.4

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

$- 6 \frac{- 8}{{(- 8)}^{2} - \sqrt{{(- 8)}^{2} - 6 \cdot - 8}}$

Step 10

Simplify the answer.

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Step 10.1

Simplify the denominator.

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Step 10.1.1

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

$- 6 \frac{- 8}{64 - \sqrt{{(- 8)}^{2} - 6 \cdot - 8}}$

Step 10.1.2

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

$- 6 \frac{- 8}{64 - \sqrt{64 - 6 \cdot - 8}}$

Step 10.1.3

Multiply $- 6$ by $- 8$ .

$- 6 \frac{- 8}{64 - \sqrt{64 + 48}}$

Step 10.1.4

Add $64$ and $48$ .

$- 6 \frac{- 8}{64 - \sqrt{112}}$

Step 10.1.5

Rewrite $112$ as $4^{2} \cdot 7$ .

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Step 10.1.5.1

Factor $16$ out of $112$ .

$- 6 \frac{- 8}{64 - \sqrt{16 (7)}}$

Step 10.1.5.2

Rewrite $16$ as $4^{2}$ .

$- 6 \frac{- 8}{64 - \sqrt{4^{2} \cdot 7}}$

Step 10.1.6

Pull terms out from under the radical.

$- 6 \frac{- 8}{64 - (4 \sqrt{7})}$

Step 10.1.7

Multiply $4$ by $- 1$ .

$- 6 \frac{- 8}{64 - 4 \sqrt{7}}$

Step 10.2

Cancel the common factor of $- 8$ and $64 - 4 \sqrt{7}$ .

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Step 10.2.1

Factor $4$ out of $- 8$ .

$- 6 \frac{4 (- 2)}{64 - 4 \sqrt{7}}$

Step 10.2.2

Cancel the common factors.

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Step 10.2.2.1

Factor $4$ out of $64$ .

$- 6 \frac{4 \cdot - 2}{4 \cdot 16 - 4 \sqrt{7}}$

Step 10.2.2.2

Factor $4$ out of $- 4 \sqrt{7}$ .

$- 6 \frac{4 \cdot - 2}{4 \cdot 16 + 4 (- \sqrt{7})}$

Step 10.2.2.3

Factor $4$ out of $4 (16) + 4 (- \sqrt{7})$ .

$- 6 \frac{4 \cdot - 2}{4 (16 - \sqrt{7})}$

Step 10.2.2.4

Cancel the common factor.

$- 6 \frac{4 \cdot - 2}{4 (16 - \sqrt{7})}$

Step 10.2.2.5

Rewrite the expression.

$- 6 \frac{- 2}{16 - \sqrt{7}}$

Step 10.3

Move the negative in front of the fraction.

$- 6 (- \frac{2}{16 - \sqrt{7}})$

Step 10.4

Multiply $\frac{2}{16 - \sqrt{7}}$ by $\frac{16 + \sqrt{7}}{16 + \sqrt{7}}$ .

$- 6 (- (\frac{2}{16 - \sqrt{7}} \cdot \frac{16 + \sqrt{7}}{16 + \sqrt{7}}))$

Step 10.5

Multiply $\frac{2}{16 - \sqrt{7}}$ by $\frac{16 + \sqrt{7}}{16 + \sqrt{7}}$ .

$- 6 (- \frac{2 (16 + \sqrt{7})}{(16 - \sqrt{7}) (16 + \sqrt{7})})$

Step 10.6

Expand the denominator using the FOIL method.

$- 6 (- \frac{2 (16 + \sqrt{7})}{256 + 16 \sqrt{7} - 16 \sqrt{7} - {\sqrt{7}}^{2}})$

Step 10.7

Simplify.

$- 6 (- \frac{2 (16 + \sqrt{7})}{249})$

Step 10.8

Cancel the common factor of $3$ .

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Step 10.8.1

Move the leading negative in $- \frac{2 (16 + \sqrt{7})}{249}$ into the numerator.

$- 6 \frac{- 2 (16 + \sqrt{7})}{249}$

Step 10.8.2

Factor $3$ out of $- 6$ .

$3 (- 2) \frac{- 2 (16 + \sqrt{7})}{249}$

Step 10.8.3

Factor $3$ out of $249$ .

$3 \cdot - 2 \frac{- 2 (16 + \sqrt{7})}{3 \cdot 83}$

Step 10.8.4

Cancel the common factor.

$3 \cdot - 2 \frac{- 2 (16 + \sqrt{7})}{3 \cdot 83}$

Step 10.8.5

Rewrite the expression.

$- 2 \frac{- 2 (16 + \sqrt{7})}{83}$

Step 10.9

Combine $- 2$ and $\frac{- 2 (16 + \sqrt{7})}{83}$ .

$\frac{- 2 (- 2 (16 + \sqrt{7}))}{83}$

Step 10.10

Multiply $- 2$ by $- 2$ .

$\frac{4 (16 + \sqrt{7})}{83}$

Step 11

The result can be shown in multiple forms.

Exact Form:

$\frac{4 (16 + \sqrt{7})}{83}$

Decimal Form:

$0.89859042 \dots$