Calculus Examples

$lim_{x \to - 8} \frac{18 x}{x - 3} - \frac{2 x}{x + 6}$

Step 1

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

$lim_{x \to - 8} \frac{18 x}{x - 3} - lim_{x \to - 8} \frac{2 x}{x + 6}$

Step 2

Move the term $18$ outside of the limit because it is constant with respect to $x$ .

$18 lim_{x \to - 8} \frac{x}{x - 3} - lim_{x \to - 8} \frac{2 x}{x + 6}$

Step 3

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

$18 \frac{lim_{x \to - 8} x}{lim_{x \to - 8} x - 3} - lim_{x \to - 8} \frac{2 x}{x + 6}$

Step 4

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

$18 \frac{lim_{x \to - 8} x}{lim_{x \to - 8} x - lim_{x \to - 8} 3} - lim_{x \to - 8} \frac{2 x}{x + 6}$

Step 5

Evaluate the limit of $3$ which is constant as $x$ approaches $- 8$ .

$18 \frac{lim_{x \to - 8} x}{lim_{x \to - 8} x - 1 \cdot 3} - lim_{x \to - 8} \frac{2 x}{x + 6}$

Step 6

Move the term $2$ outside of the limit because it is constant with respect to $x$ .

$18 \frac{lim_{x \to - 8} x}{lim_{x \to - 8} x - 1 \cdot 3} - 2 lim_{x \to - 8} \frac{x}{x + 6}$

Step 7

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

$18 \frac{lim_{x \to - 8} x}{lim_{x \to - 8} x - 1 \cdot 3} - 2 \frac{lim_{x \to - 8} x}{lim_{x \to - 8} x + 6}$

Step 8

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

$18 \frac{lim_{x \to - 8} x}{lim_{x \to - 8} x - 1 \cdot 3} - 2 \frac{lim_{x \to - 8} x}{lim_{x \to - 8} x + lim_{x \to - 8} 6}$

Step 9

Evaluate the limit of $6$ which is constant as $x$ approaches $- 8$ .

$18 \frac{lim_{x \to - 8} x}{lim_{x \to - 8} x - 1 \cdot 3} - 2 \frac{lim_{x \to - 8} x}{lim_{x \to - 8} x + 6}$

Step 10

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

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Step 10.1

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

$18 \frac{- 8}{lim_{x \to - 8} x - 1 \cdot 3} - 2 \frac{lim_{x \to - 8} x}{lim_{x \to - 8} x + 6}$

Step 10.2

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

$18 \frac{- 8}{- 8 - 1 \cdot 3} - 2 \frac{lim_{x \to - 8} x}{lim_{x \to - 8} x + 6}$

Step 10.3

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

$18 \frac{- 8}{- 8 - 1 \cdot 3} - 2 \frac{- 8}{lim_{x \to - 8} x + 6}$

Step 10.4

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

$18 \frac{- 8}{- 8 - 1 \cdot 3} - 2 \frac{- 8}{- 8 + 6}$

Step 11

Simplify the answer.

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Step 11.1

Simplify each term.

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Step 11.1.1

Simplify the denominator.

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Step 11.1.1.1

Multiply $- 1$ by $3$ .

$18 \frac{- 8}{- 8 - 3} - 2 \frac{- 8}{- 8 + 6}$

Step 11.1.1.2

Subtract $3$ from $- 8$ .

$18 (\frac{- 8}{- 11}) - 2 \frac{- 8}{- 8 + 6}$

Step 11.1.2

Dividing two negative values results in a positive value.

$18 (\frac{8}{11}) - 2 \frac{- 8}{- 8 + 6}$

Step 11.1.3

Multiply $18 (\frac{8}{11})$ .

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Step 11.1.3.1

Combine $18$ and $\frac{8}{11}$ .

$\frac{18 \cdot 8}{11} - 2 \frac{- 8}{- 8 + 6}$

Step 11.1.3.2

Multiply $18$ by $8$ .

$\frac{144}{11} - 2 \frac{- 8}{- 8 + 6}$

Step 11.1.4

Cancel the common factor of $- 8$ and $- 8 + 6$ .

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Step 11.1.4.1

Factor $2$ out of $- 8$ .

$\frac{144}{11} - 2 \frac{2 (- 4)}{- 8 + 6}$

Step 11.1.4.2

Cancel the common factors.

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Step 11.1.4.2.1

Factor $2$ out of $- 8$ .

$\frac{144}{11} - 2 \frac{2 \cdot - 4}{2 \cdot - 4 + 6}$

Step 11.1.4.2.2

Factor $2$ out of $6$ .

$\frac{144}{11} - 2 \frac{2 (- 4)}{2 \cdot - 4 + 2 \cdot 3}$

Step 11.1.4.2.3

Factor $2$ out of $2 \cdot - 4 + 2 \cdot 3$ .

$\frac{144}{11} - 2 \frac{2 (- 4)}{2 \cdot (- 4 + 3)}$

Step 11.1.4.2.4

Cancel the common factor.

$\frac{144}{11} - 2 \frac{2 \cdot - 4}{2 \cdot (- 4 + 3)}$

Step 11.1.4.2.5

Rewrite the expression.

$\frac{144}{11} - 2 \frac{- 4}{- 4 + 3}$

Step 11.1.5

Add $- 4$ and $3$ .

$\frac{144}{11} - 2 (\frac{- 4}{- 1})$

Step 11.1.6

Divide $- 4$ by $- 1$ .

$\frac{144}{11} - 2 \cdot 4$

Step 11.1.7

Multiply $- 2$ by $4$ .

$\frac{144}{11} - 8$

Step 11.2

To write $- 8$ as a fraction with a common denominator, multiply by $\frac{11}{11}$ .

$\frac{144}{11} - 8 \cdot \frac{11}{11}$

Step 11.3

Combine $- 8$ and $\frac{11}{11}$ .

$\frac{144}{11} + \frac{- 8 \cdot 11}{11}$

Step 11.4

Combine the numerators over the common denominator.

$\frac{144 - 8 \cdot 11}{11}$

Step 11.5

Simplify the numerator.

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Step 11.5.1

Multiply $- 8$ by $11$ .

$\frac{144 - 88}{11}$

Step 11.5.2

Subtract $88$ from $144$ .

$\frac{56}{11}$

Step 12

The result can be shown in multiple forms.

Exact Form:

$\frac{56}{11}$

Decimal Form:

$5.\overline{09}$