Calculus Examples

$lim_{x \to - \infty} \frac{- 2 x^{3} + x^{2} + 4 x - 4}{- 2 x^{3} - 2 x^{2} - 7 x - 7}$

Step 1

Divide the numerator and denominator by the highest power of $x$ in the denominator, which is $x^{3}$ .

$lim_{x \to - \infty} \frac{\frac{- 2 x^{3}}{x^{3}} + \frac{x^{2}}{x^{3}} + \frac{4 x}{x^{3}} + \frac{- 4}{x^{3}}}{\frac{- 2 x^{3}}{x^{3}} + \frac{- 2 x^{2}}{x^{3}} + \frac{- 7 x}{x^{3}} + \frac{- 7}{x^{3}}}$

Step 2

Evaluate the limit.

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Step 2.1

Simplify each term.

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Step 2.1.1

Cancel the common factor of $x^{3}$ .

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Step 2.1.1.1

Cancel the common factor.

Step 2.1.1.2

Divide $- 2$ by $1$ .

$lim_{x \to - \infty} \frac{- 2 + \frac{x^{2}}{x^{3}} + \frac{4 x}{x^{3}} + \frac{- 4}{x^{3}}}{\frac{- 2 x^{3}}{x^{3}} + \frac{- 2 x^{2}}{x^{3}} + \frac{- 7 x}{x^{3}} + \frac{- 7}{x^{3}}}$

Step 2.1.2

Cancel the common factor of $x^{2}$ and $x^{3}$ .

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Step 2.1.2.1

Multiply by $1$ .

$lim_{x \to - \infty} \frac{- 2 + \frac{x^{2} \cdot 1}{x^{3}} + \frac{4 x}{x^{3}} + \frac{- 4}{x^{3}}}{\frac{- 2 x^{3}}{x^{3}} + \frac{- 2 x^{2}}{x^{3}} + \frac{- 7 x}{x^{3}} + \frac{- 7}{x^{3}}}$

Step 2.1.2.2

Cancel the common factors.

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Step 2.1.2.2.1

Factor $x^{2}$ out of $x^{3}$ .

$lim_{x \to - \infty} \frac{- 2 + \frac{x^{2} \cdot 1}{x^{2} x} + \frac{4 x}{x^{3}} + \frac{- 4}{x^{3}}}{\frac{- 2 x^{3}}{x^{3}} + \frac{- 2 x^{2}}{x^{3}} + \frac{- 7 x}{x^{3}} + \frac{- 7}{x^{3}}}$

Step 2.1.2.2.2

Cancel the common factor.

$lim_{x \to - \infty} \frac{- 2 + \frac{x^{2} \cdot 1}{x^{2} x} + \frac{4 x}{x^{3}} + \frac{- 4}{x^{3}}}{\frac{- 2 x^{3}}{x^{3}} + \frac{- 2 x^{2}}{x^{3}} + \frac{- 7 x}{x^{3}} + \frac{- 7}{x^{3}}}$

Step 2.1.2.2.3

Rewrite the expression.

$lim_{x \to - \infty} \frac{- 2 + \frac{1}{x} + \frac{4 x}{x^{3}} + \frac{- 4}{x^{3}}}{\frac{- 2 x^{3}}{x^{3}} + \frac{- 2 x^{2}}{x^{3}} + \frac{- 7 x}{x^{3}} + \frac{- 7}{x^{3}}}$

Step 2.1.3

Cancel the common factor of $x$ and $x^{3}$ .

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Step 2.1.3.1

Factor $x$ out of $4 x$ .

$lim_{x \to - \infty} \frac{- 2 + \frac{1}{x} + \frac{x \cdot 4}{x^{3}} + \frac{- 4}{x^{3}}}{\frac{- 2 x^{3}}{x^{3}} + \frac{- 2 x^{2}}{x^{3}} + \frac{- 7 x}{x^{3}} + \frac{- 7}{x^{3}}}$

Step 2.1.3.2

Cancel the common factors.

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Step 2.1.3.2.1

Factor $x$ out of $x^{3}$ .

$lim_{x \to - \infty} \frac{- 2 + \frac{1}{x} + \frac{x \cdot 4}{x \cdot x^{2}} + \frac{- 4}{x^{3}}}{\frac{- 2 x^{3}}{x^{3}} + \frac{- 2 x^{2}}{x^{3}} + \frac{- 7 x}{x^{3}} + \frac{- 7}{x^{3}}}$

Step 2.1.3.2.2

Cancel the common factor.

$lim_{x \to - \infty} \frac{- 2 + \frac{1}{x} + \frac{x \cdot 4}{x \cdot x^{2}} + \frac{- 4}{x^{3}}}{\frac{- 2 x^{3}}{x^{3}} + \frac{- 2 x^{2}}{x^{3}} + \frac{- 7 x}{x^{3}} + \frac{- 7}{x^{3}}}$

Step 2.1.3.2.3

Rewrite the expression.

$lim_{x \to - \infty} \frac{- 2 + \frac{1}{x} + \frac{4}{x^{2}} + \frac{- 4}{x^{3}}}{\frac{- 2 x^{3}}{x^{3}} + \frac{- 2 x^{2}}{x^{3}} + \frac{- 7 x}{x^{3}} + \frac{- 7}{x^{3}}}$

Step 2.1.4

Move the negative in front of the fraction.

$lim_{x \to - \infty} \frac{- 2 + \frac{1}{x} + \frac{4}{x^{2}} - \frac{4}{x^{3}}}{\frac{- 2 x^{3}}{x^{3}} + \frac{- 2 x^{2}}{x^{3}} + \frac{- 7 x}{x^{3}} + \frac{- 7}{x^{3}}}$

Step 2.2

Simplify each term.

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Step 2.2.1

Cancel the common factor of $x^{3}$ .

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Step 2.2.1.1

Cancel the common factor.

$lim_{x \to - \infty} \frac{- 2 + \frac{1}{x} + \frac{4}{x^{2}} - \frac{4}{x^{3}}}{\frac{- 2 x^{3}}{x^{3}} + \frac{- 2 x^{2}}{x^{3}} + \frac{- 7 x}{x^{3}} + \frac{- 7}{x^{3}}}$

Step 2.2.1.2

Divide $- 2$ by $1$ .

$lim_{x \to - \infty} \frac{- 2 + \frac{1}{x} + \frac{4}{x^{2}} - \frac{4}{x^{3}}}{- 2 + \frac{- 2 x^{2}}{x^{3}} + \frac{- 7 x}{x^{3}} + \frac{- 7}{x^{3}}}$

Step 2.2.2

Cancel the common factor of $x^{2}$ and $x^{3}$ .

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Step 2.2.2.1

Factor $x^{2}$ out of $- 2 x^{2}$ .

$lim_{x \to - \infty} \frac{- 2 + \frac{1}{x} + \frac{4}{x^{2}} - \frac{4}{x^{3}}}{- 2 + \frac{x^{2} \cdot - 2}{x^{3}} + \frac{- 7 x}{x^{3}} + \frac{- 7}{x^{3}}}$

Step 2.2.2.2

Cancel the common factors.

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Step 2.2.2.2.1

Factor $x^{2}$ out of $x^{3}$ .

$lim_{x \to - \infty} \frac{- 2 + \frac{1}{x} + \frac{4}{x^{2}} - \frac{4}{x^{3}}}{- 2 + \frac{x^{2} \cdot - 2}{x^{2} x} + \frac{- 7 x}{x^{3}} + \frac{- 7}{x^{3}}}$

Step 2.2.2.2.2

Cancel the common factor.

$lim_{x \to - \infty} \frac{- 2 + \frac{1}{x} + \frac{4}{x^{2}} - \frac{4}{x^{3}}}{- 2 + \frac{x^{2} \cdot - 2}{x^{2} x} + \frac{- 7 x}{x^{3}} + \frac{- 7}{x^{3}}}$

Step 2.2.2.2.3

Rewrite the expression.

$lim_{x \to - \infty} \frac{- 2 + \frac{1}{x} + \frac{4}{x^{2}} - \frac{4}{x^{3}}}{- 2 + \frac{- 2}{x} + \frac{- 7 x}{x^{3}} + \frac{- 7}{x^{3}}}$

Step 2.2.3

Move the negative in front of the fraction.

$lim_{x \to - \infty} \frac{- 2 + \frac{1}{x} + \frac{4}{x^{2}} - \frac{4}{x^{3}}}{- 2 - \frac{2}{x} + \frac{- 7 x}{x^{3}} + \frac{- 7}{x^{3}}}$

Step 2.2.4

Cancel the common factor of $x$ and $x^{3}$ .

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Step 2.2.4.1

Factor $x$ out of $- 7 x$ .

$lim_{x \to - \infty} \frac{- 2 + \frac{1}{x} + \frac{4}{x^{2}} - \frac{4}{x^{3}}}{- 2 - \frac{2}{x} + \frac{x \cdot - 7}{x^{3}} + \frac{- 7}{x^{3}}}$

Step 2.2.4.2

Cancel the common factors.

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Step 2.2.4.2.1

Factor $x$ out of $x^{3}$ .

$lim_{x \to - \infty} \frac{- 2 + \frac{1}{x} + \frac{4}{x^{2}} - \frac{4}{x^{3}}}{- 2 - \frac{2}{x} + \frac{x \cdot - 7}{x \cdot x^{2}} + \frac{- 7}{x^{3}}}$

Step 2.2.4.2.2

Cancel the common factor.

$lim_{x \to - \infty} \frac{- 2 + \frac{1}{x} + \frac{4}{x^{2}} - \frac{4}{x^{3}}}{- 2 - \frac{2}{x} + \frac{x \cdot - 7}{x \cdot x^{2}} + \frac{- 7}{x^{3}}}$

Step 2.2.4.2.3

Rewrite the expression.

$lim_{x \to - \infty} \frac{- 2 + \frac{1}{x} + \frac{4}{x^{2}} - \frac{4}{x^{3}}}{- 2 - \frac{2}{x} + \frac{- 7}{x^{2}} + \frac{- 7}{x^{3}}}$

Step 2.2.5

Move the negative in front of the fraction.

$lim_{x \to - \infty} \frac{- 2 + \frac{1}{x} + \frac{4}{x^{2}} - \frac{4}{x^{3}}}{- 2 - \frac{2}{x} - \frac{7}{x^{2}} + \frac{- 7}{x^{3}}}$

Step 2.2.6

Move the negative in front of the fraction.

$lim_{x \to - \infty} \frac{- 2 + \frac{1}{x} + \frac{4}{x^{2}} - \frac{4}{x^{3}}}{- 2 - \frac{2}{x} - \frac{7}{x^{2}} - \frac{7}{x^{3}}}$

Step 2.3

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

$\frac{lim_{x \to - \infty} - 2 + \frac{1}{x} + \frac{4}{x^{2}} - \frac{4}{x^{3}}}{lim_{x \to - \infty} - 2 - \frac{2}{x} - \frac{7}{x^{2}} - \frac{7}{x^{3}}}$

Step 2.4

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

$\frac{- lim_{x \to - \infty} 2 + lim_{x \to - \infty} \frac{1}{x} + lim_{x \to - \infty} \frac{4}{x^{2}} - lim_{x \to - \infty} \frac{4}{x^{3}}}{lim_{x \to - \infty} - 2 - \frac{2}{x} - \frac{7}{x^{2}} - \frac{7}{x^{3}}}$

Step 2.5

Evaluate the limit of $2$ which is constant as $x$ approaches $- \infty$ .

$\frac{- 1 \cdot 2 + lim_{x \to - \infty} \frac{1}{x} + lim_{x \to - \infty} \frac{4}{x^{2}} - lim_{x \to - \infty} \frac{4}{x^{3}}}{lim_{x \to - \infty} - 2 - \frac{2}{x} - \frac{7}{x^{2}} - \frac{7}{x^{3}}}$

Step 3

Since its numerator approaches a real number while its denominator is unbounded, the fraction $\frac{1}{x}$ approaches $0$ .

$\frac{- 2 + 0 + lim_{x \to - \infty} \frac{4}{x^{2}} - lim_{x \to - \infty} \frac{4}{x^{3}}}{lim_{x \to - \infty} - 2 - \frac{2}{x} - \frac{7}{x^{2}} - \frac{7}{x^{3}}}$

Step 4

Move the term $4$ outside of the limit because it is constant with respect to $x$ .

$\frac{- 2 + 0 + 4 lim_{x \to - \infty} \frac{1}{x^{2}} - lim_{x \to - \infty} \frac{4}{x^{3}}}{lim_{x \to - \infty} - 2 - \frac{2}{x} - \frac{7}{x^{2}} - \frac{7}{x^{3}}}$

Step 5

Since its numerator approaches a real number while its denominator is unbounded, the fraction $\frac{1}{x^{2}}$ approaches $0$ .

$\frac{- 2 + 0 + 4 \cdot 0 - lim_{x \to - \infty} \frac{4}{x^{3}}}{lim_{x \to - \infty} - 2 - \frac{2}{x} - \frac{7}{x^{2}} - \frac{7}{x^{3}}}$

Step 6

Move the term $4$ outside of the limit because it is constant with respect to $x$ .

$\frac{- 2 + 0 + 4 \cdot 0 - 4 lim_{x \to - \infty} \frac{1}{x^{3}}}{lim_{x \to - \infty} - 2 - \frac{2}{x} - \frac{7}{x^{2}} - \frac{7}{x^{3}}}$

Step 7

Since its numerator approaches a real number while its denominator is unbounded, the fraction $\frac{1}{x^{3}}$ approaches $0$ .

$\frac{- 2 + 0 + 4 \cdot 0 - 4 \cdot 0}{lim_{x \to - \infty} - 2 - \frac{2}{x} - \frac{7}{x^{2}} - \frac{7}{x^{3}}}$

Step 8

Evaluate the limit.

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Step 8.1

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

$\frac{- 2 + 0 + 4 \cdot 0 - 4 \cdot 0}{- lim_{x \to - \infty} 2 - lim_{x \to - \infty} \frac{2}{x} - lim_{x \to - \infty} \frac{7}{x^{2}} - lim_{x \to - \infty} \frac{7}{x^{3}}}$

Step 8.2

Evaluate the limit of $2$ which is constant as $x$ approaches $- \infty$ .

$\frac{- 2 + 0 + 4 \cdot 0 - 4 \cdot 0}{- 1 \cdot 2 - lim_{x \to - \infty} \frac{2}{x} - lim_{x \to - \infty} \frac{7}{x^{2}} - lim_{x \to - \infty} \frac{7}{x^{3}}}$

Step 8.3

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

$\frac{- 2 + 0 + 4 \cdot 0 - 4 \cdot 0}{- 2 - 2 lim_{x \to - \infty} \frac{1}{x} - lim_{x \to - \infty} \frac{7}{x^{2}} - lim_{x \to - \infty} \frac{7}{x^{3}}}$

Step 9

Since its numerator approaches a real number while its denominator is unbounded, the fraction $\frac{1}{x}$ approaches $0$ .

$\frac{- 2 + 0 + 4 \cdot 0 - 4 \cdot 0}{- 2 - 2 \cdot 0 - lim_{x \to - \infty} \frac{7}{x^{2}} - lim_{x \to - \infty} \frac{7}{x^{3}}}$

Step 10

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

$\frac{- 2 + 0 + 4 \cdot 0 - 4 \cdot 0}{- 2 - 2 \cdot 0 - 7 lim_{x \to - \infty} \frac{1}{x^{2}} - lim_{x \to - \infty} \frac{7}{x^{3}}}$

Step 11

Since its numerator approaches a real number while its denominator is unbounded, the fraction $\frac{1}{x^{2}}$ approaches $0$ .

$\frac{- 2 + 0 + 4 \cdot 0 - 4 \cdot 0}{- 2 - 2 \cdot 0 - 7 \cdot 0 - lim_{x \to - \infty} \frac{7}{x^{3}}}$

Step 12

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

$\frac{- 2 + 0 + 4 \cdot 0 - 4 \cdot 0}{- 2 - 2 \cdot 0 - 7 \cdot 0 - 7 lim_{x \to - \infty} \frac{1}{x^{3}}}$

Step 13

Since its numerator approaches a real number while its denominator is unbounded, the fraction $\frac{1}{x^{3}}$ approaches $0$ .

$\frac{- 2 + 0 + 4 \cdot 0 - 4 \cdot 0}{- 2 - 2 \cdot 0 - 7 \cdot 0 - 7 \cdot 0}$

Step 14

Simplify the answer.

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Step 14.1

Cancel the common factor of $- 2 + 0 + 4 \cdot 0 - 4 \cdot 0$ and $- 2 - 2 \cdot 0 - 7 \cdot 0 - 7 \cdot 0$ .

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Step 14.1.1

Rewrite $- 2$ as $- 1 (2)$ .

$\frac{- 2 + 0 + 4 \cdot 0 - 4 \cdot 0}{- 1 (2) - 2 \cdot 0 - 7 \cdot 0 - 7 \cdot 0}$

Step 14.1.2

Factor $- 1$ out of $- 2 \cdot 0$ .

$\frac{- 2 + 0 + 4 \cdot 0 - 4 \cdot 0}{- 1 (2) - (2 \cdot 0) - 7 \cdot 0 - 7 \cdot 0}$

Step 14.1.3

Factor $- 1$ out of $- 1 (2) - (2 \cdot 0)$ .

$\frac{- 2 + 0 + 4 \cdot 0 - 4 \cdot 0}{- 1 (2 + 2 \cdot 0) - 7 \cdot 0 - 7 \cdot 0}$

Step 14.1.4

Factor $- 1$ out of $- 7 \cdot 0$ .

$\frac{- 2 + 0 + 4 \cdot 0 - 4 \cdot 0}{- 1 (2 + 2 \cdot 0) - (7 \cdot 0) - 7 \cdot 0}$

Step 14.1.5

Factor $- 1$ out of $- 1 (2 + 2 \cdot 0) - (7 \cdot 0)$ .

$\frac{- 2 + 0 + 4 \cdot 0 - 4 \cdot 0}{- 1 (2 + 2 \cdot 0 + 7 \cdot 0) - 7 \cdot 0}$

Step 14.1.6

Factor $- 1$ out of $- 7 \cdot 0$ .

$\frac{- 2 + 0 + 4 \cdot 0 - 4 \cdot 0}{- 1 (2 + 2 \cdot 0 + 7 \cdot 0) - (7 \cdot 0)}$

Step 14.1.7

Factor $- 1$ out of $- 1 (2 + 2 \cdot 0 + 7 \cdot 0) - (7 \cdot 0)$ .

$\frac{- 2 + 0 + 4 \cdot 0 - 4 \cdot 0}{- 1 (2 + 2 \cdot 0 + 7 \cdot 0 + 7 \cdot 0)}$

Step 14.1.8

Reorder terms.

$\frac{- 2 + 0 + 4 \cdot 0 - 4 \cdot 0}{- 1 (2 + 0 \cdot 2 + 0 \cdot 7 + 0 \cdot 7)}$

Step 14.1.9

Factor $2$ out of $- 2$ .

$\frac{2 (- 1) + 0 + 4 \cdot 0 - 4 \cdot 0}{- 1 (2 + 0 \cdot 2 + 0 \cdot 7 + 0 \cdot 7)}$

Step 14.1.10

Factor $2$ out of $0$ .

$\frac{2 \cdot - 1 + 2 \cdot 0 + 4 \cdot 0 - 4 \cdot 0}{- 1 (2 + 0 \cdot 2 + 0 \cdot 7 + 0 \cdot 7)}$

Step 14.1.11

Factor $2$ out of $2 \cdot - 1 + 2 \cdot 0$ .

$\frac{2 \cdot (- 1 + 0) + 4 \cdot 0 - 4 \cdot 0}{- 1 (2 + 0 \cdot 2 + 0 \cdot 7 + 0 \cdot 7)}$

Step 14.1.12

Factor $2$ out of $4 \cdot 0$ .

$\frac{2 \cdot (- 1 + 0) + 2 (2 \cdot 0) - 4 \cdot 0}{- 1 (2 + 0 \cdot 2 + 0 \cdot 7 + 0 \cdot 7)}$

Step 14.1.13

Factor $2$ out of $2 \cdot (- 1 + 0) + 2 (2 \cdot 0)$ .

$\frac{2 \cdot (- 1 + 0 + 2 \cdot 0) - 4 \cdot 0}{- 1 (2 + 0 \cdot 2 + 0 \cdot 7 + 0 \cdot 7)}$

Step 14.1.14

Factor $2$ out of $- 4 \cdot 0$ .

$\frac{2 \cdot (- 1 + 0 + 2 \cdot 0) + 2 (- 2 \cdot 0)}{- 1 (2 + 0 \cdot 2 + 0 \cdot 7 + 0 \cdot 7)}$

Step 14.1.15

Factor $2$ out of $2 \cdot (- 1 + 0 + 2 \cdot 0) + 2 (- 2 \cdot 0)$ .

$\frac{2 \cdot (- 1 + 0 + 2 \cdot 0 - 2 \cdot 0)}{- 1 (2 + 0 \cdot 2 + 0 \cdot 7 + 0 \cdot 7)}$

Step 14.1.16

Cancel the common factors.

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Step 14.1.16.1

Factor $2$ out of $- 1 (2 + 0 \cdot 2 + 0 \cdot 7 + 0 \cdot 7)$ .

$\frac{2 \cdot (- 1 + 0 + 2 \cdot 0 - 2 \cdot 0)}{2 (- 1 (1 + 0 \cdot 2 + 0 \cdot 7 + 0 \cdot 7))}$

Step 14.1.16.2

Cancel the common factor.

$\frac{2 \cdot (- 1 + 0 + 2 \cdot 0 - 2 \cdot 0)}{2 (- 1 (1 + 0 \cdot 2 + 0 \cdot 7 + 0 \cdot 7))}$

Step 14.1.16.3

Rewrite the expression.

$\frac{- 1 + 0 + 2 \cdot 0 - 2 \cdot 0}{- 1 (1 + 0 \cdot 2 + 0 \cdot 7 + 0 \cdot 7)}$

Step 14.2

Simplify the numerator.

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Step 14.2.1

Multiply $2$ by $0$ .

$\frac{- 1 + 0 + 0 - 2 \cdot 0}{- 1 (1 + 0 \cdot 2 + 0 \cdot 7 + 0 \cdot 7)}$

Step 14.2.2

Multiply $- 2$ by $0$ .

$\frac{- 1 + 0 + 0 + 0}{- 1 (1 + 0 \cdot 2 + 0 \cdot 7 + 0 \cdot 7)}$

Step 14.2.3

Add $- 1 + 0 + 0$ and $0$ .

$\frac{- 1 + 0 + 0}{- 1 (1 + 0 \cdot 2 + 0 \cdot 7 + 0 \cdot 7)}$

Step 14.2.4

Add $- 1 + 0$ and $0$ .

$\frac{- 1 + 0}{- 1 (1 + 0 \cdot 2 + 0 \cdot 7 + 0 \cdot 7)}$

Step 14.2.5

Add $- 1$ and $0$ .

$\frac{- 1}{- 1 (1 + 0 \cdot 2 + 0 \cdot 7 + 0 \cdot 7)}$

Step 14.3

Simplify the denominator.

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Step 14.3.1

Multiply $0$ by $2$ .

$\frac{- 1}{- 1 (1 + 0 + 0 \cdot 7 + 0 \cdot 7)}$

Step 14.3.2

Multiply $0$ by $7$ .

$\frac{- 1}{- 1 (1 + 0 + 0 + 0 \cdot 7)}$

Step 14.3.3

Multiply $0$ by $7$ .

$\frac{- 1}{- 1 (1 + 0 + 0 + 0)}$

Step 14.3.4

Add $1 + 0 + 0$ and $0$ .

$\frac{- 1}{- 1 (1 + 0 + 0)}$

Step 14.3.5

Add $1 + 0$ and $0$ .

$\frac{- 1}{- 1 (1 + 0)}$

Step 14.3.6

Add $1$ and $0$ .

$\frac{- 1}{- 1 \cdot 1}$

Step 14.4

Multiply $- 1$ by $1$ .

$\frac{- 1}{- 1}$

Step 14.5

Divide $- 1$ by $- 1$ .

$1$