Calculus Examples

$lim_{x \to - 8} \frac{7 x - 6}{x}$

Step 1

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

$\frac{lim_{x \to - 8} 7 x - 6}{lim_{x \to - 8} x}$

Step 2

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

$\frac{lim_{x \to - 8} 7 x - lim_{x \to - 8} 6}{lim_{x \to - 8} x}$

Step 3

Move the term $7$ outside of the limit because it is constant with respect to $x$ .

$\frac{7 lim_{x \to - 8} x - lim_{x \to - 8} 6}{lim_{x \to - 8} x}$

Step 4

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

$\frac{7 lim_{x \to - 8} x - 1 \cdot 6}{lim_{x \to - 8} x}$

Step 5

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

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Step 5.1

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

$\frac{7 \cdot - 8 - 1 \cdot 6}{lim_{x \to - 8} x}$

Step 5.2

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

$\frac{7 \cdot - 8 - 1 \cdot 6}{- 8}$

Step 6

Simplify the answer.

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Step 6.1

Cancel the common factor of $7 \cdot - 8 - 1 \cdot 6$ and $- 8$ .

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Step 6.1.1

Reorder terms.

$\frac{- 1 \cdot 6 - 8 \cdot 7}{- 8}$

Step 6.1.2

Factor $2$ out of $- 1 \cdot 6$ .

$\frac{2 (- 1 \cdot 3) - 8 \cdot 7}{- 8}$

Step 6.1.3

Factor $2$ out of $- 8 \cdot 7$ .

$\frac{2 (- 1 \cdot 3) + 2 (- 4 \cdot 7)}{- 8}$

Step 6.1.4

Factor $2$ out of $2 (- 1 \cdot 3) + 2 (- 4 \cdot 7)$ .

$\frac{2 (- 1 \cdot 3 - 4 \cdot 7)}{- 8}$

Step 6.1.5

Cancel the common factors.

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Step 6.1.5.1

Factor $2$ out of $- 8$ .

$\frac{2 (- 1 \cdot 3 - 4 \cdot 7)}{2 (- 4)}$

Step 6.1.5.2

Cancel the common factor.

$\frac{2 (- 1 \cdot 3 - 4 \cdot 7)}{2 \cdot - 4}$

Step 6.1.5.3

Rewrite the expression.

$\frac{- 1 \cdot 3 - 4 \cdot 7}{- 4}$

Step 6.2

Simplify the numerator.

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Step 6.2.1

Multiply $- 1$ by $3$ .

$\frac{- 3 - 4 \cdot 7}{- 4}$

Step 6.2.2

Multiply $- 4$ by $7$ .

$\frac{- 3 - 28}{- 4}$

Step 6.2.3

Subtract $28$ from $- 3$ .

$\frac{- 31}{- 4}$

Step 6.3

Dividing two negative values results in a positive value.

$\frac{31}{4}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$\frac{31}{4}$

Decimal Form:

$7.75$