Calculus Examples

$lim_{x \to - \infty} \frac{- 6 x^{3} + 7 x + 7}{8 x^{3} - 8 x + 5}$

Step 1

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

$lim_{x \to - \infty} \frac{\frac{- 6 x^{3}}{x^{3}} + \frac{7 x}{x^{3}} + \frac{7}{x^{3}}}{\frac{8 x^{3}}{x^{3}} + \frac{- 8 x}{x^{3}} + \frac{5}{x^{3}}}$

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.

$lim_{x \to - \infty} \frac{\frac{- 6 x^{3}}{x^{3}} + \frac{7 x}{x^{3}} + \frac{7}{x^{3}}}{\frac{8 x^{3}}{x^{3}} + \frac{- 8 x}{x^{3}} + \frac{5}{x^{3}}}$

Step 2.1.1.2

Divide $- 6$ by $1$ .

$lim_{x \to - \infty} \frac{- 6 + \frac{7 x}{x^{3}} + \frac{7}{x^{3}}}{\frac{8 x^{3}}{x^{3}} + \frac{- 8 x}{x^{3}} + \frac{5}{x^{3}}}$

Step 2.1.2

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

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Step 2.1.2.1

Factor $x$ out of $7 x$ .

$lim_{x \to - \infty} \frac{- 6 + \frac{x \cdot 7}{x^{3}} + \frac{7}{x^{3}}}{\frac{8 x^{3}}{x^{3}} + \frac{- 8 x}{x^{3}} + \frac{5}{x^{3}}}$

Step 2.1.2.2

Cancel the common factors.

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Step 2.1.2.2.1

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

$lim_{x \to - \infty} \frac{- 6 + \frac{x \cdot 7}{x \cdot x^{2}} + \frac{7}{x^{3}}}{\frac{8 x^{3}}{x^{3}} + \frac{- 8 x}{x^{3}} + \frac{5}{x^{3}}}$

Step 2.1.2.2.2

Cancel the common factor.

$lim_{x \to - \infty} \frac{- 6 + \frac{x \cdot 7}{x \cdot x^{2}} + \frac{7}{x^{3}}}{\frac{8 x^{3}}{x^{3}} + \frac{- 8 x}{x^{3}} + \frac{5}{x^{3}}}$

Step 2.1.2.2.3

Rewrite the expression.

$lim_{x \to - \infty} \frac{- 6 + \frac{7}{x^{2}} + \frac{7}{x^{3}}}{\frac{8 x^{3}}{x^{3}} + \frac{- 8 x}{x^{3}} + \frac{5}{x^{3}}}$

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.

$lim_{x \to - \infty} \frac{- 6 + \frac{7}{x^{2}} + \frac{7}{x^{3}}}{\frac{8 x^{3}}{x^{3}} + \frac{- 8 x}{x^{3}} + \frac{5}{x^{3}}}$

Step 2.2.1.2

Divide $8$ by $1$ .

$lim_{x \to - \infty} \frac{- 6 + \frac{7}{x^{2}} + \frac{7}{x^{3}}}{8 + \frac{- 8 x}{x^{3}} + \frac{5}{x^{3}}}$

Step 2.2.2

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

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Step 2.2.2.1

Factor $x$ out of $- 8 x$ .

$lim_{x \to - \infty} \frac{- 6 + \frac{7}{x^{2}} + \frac{7}{x^{3}}}{8 + \frac{x \cdot - 8}{x^{3}} + \frac{5}{x^{3}}}$

Step 2.2.2.2

Cancel the common factors.

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Step 2.2.2.2.1

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

$lim_{x \to - \infty} \frac{- 6 + \frac{7}{x^{2}} + \frac{7}{x^{3}}}{8 + \frac{x \cdot - 8}{x \cdot x^{2}} + \frac{5}{x^{3}}}$

Step 2.2.2.2.2

Cancel the common factor.

$lim_{x \to - \infty} \frac{- 6 + \frac{7}{x^{2}} + \frac{7}{x^{3}}}{8 + \frac{x \cdot - 8}{x \cdot x^{2}} + \frac{5}{x^{3}}}$

Step 2.2.2.2.3

Rewrite the expression.

$lim_{x \to - \infty} \frac{- 6 + \frac{7}{x^{2}} + \frac{7}{x^{3}}}{8 + \frac{- 8}{x^{2}} + \frac{5}{x^{3}}}$

Step 2.2.3

Move the negative in front of the fraction.

$lim_{x \to - \infty} \frac{- 6 + \frac{7}{x^{2}} + \frac{7}{x^{3}}}{8 - \frac{8}{x^{2}} + \frac{5}{x^{3}}}$

Step 2.3

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

$\frac{lim_{x \to - \infty} - 6 + \frac{7}{x^{2}} + \frac{7}{x^{3}}}{lim_{x \to - \infty} 8 - \frac{8}{x^{2}} + \frac{5}{x^{3}}}$

Step 2.4

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

$\frac{- lim_{x \to - \infty} 6 + lim_{x \to - \infty} \frac{7}{x^{2}} + lim_{x \to - \infty} \frac{7}{x^{3}}}{lim_{x \to - \infty} 8 - \frac{8}{x^{2}} + \frac{5}{x^{3}}}$

Step 2.5

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

$\frac{- 1 \cdot 6 + lim_{x \to - \infty} \frac{7}{x^{2}} + lim_{x \to - \infty} \frac{7}{x^{3}}}{lim_{x \to - \infty} 8 - \frac{8}{x^{2}} + \frac{5}{x^{3}}}$

Step 2.6

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

$\frac{- 6 + 7 lim_{x \to - \infty} \frac{1}{x^{2}} + lim_{x \to - \infty} \frac{7}{x^{3}}}{lim_{x \to - \infty} 8 - \frac{8}{x^{2}} + \frac{5}{x^{3}}}$

Step 3

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

$\frac{- 6 + 7 \cdot 0 + lim_{x \to - \infty} \frac{7}{x^{3}}}{lim_{x \to - \infty} 8 - \frac{8}{x^{2}} + \frac{5}{x^{3}}}$

Step 4

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

$\frac{- 6 + 7 \cdot 0 + 7 lim_{x \to - \infty} \frac{1}{x^{3}}}{lim_{x \to - \infty} 8 - \frac{8}{x^{2}} + \frac{5}{x^{3}}}$

Step 5

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

$\frac{- 6 + 7 \cdot 0 + 7 \cdot 0}{lim_{x \to - \infty} 8 - \frac{8}{x^{2}} + \frac{5}{x^{3}}}$

Step 6

Evaluate the limit.

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Step 6.1

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

$\frac{- 6 + 7 \cdot 0 + 7 \cdot 0}{lim_{x \to - \infty} 8 - lim_{x \to - \infty} \frac{8}{x^{2}} + lim_{x \to - \infty} \frac{5}{x^{3}}}$

Step 6.2

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

$\frac{- 6 + 7 \cdot 0 + 7 \cdot 0}{8 - lim_{x \to - \infty} \frac{8}{x^{2}} + lim_{x \to - \infty} \frac{5}{x^{3}}}$

Step 6.3

Move the term $8$ outside of the limit because it is constant with respect to $x$ .

$\frac{- 6 + 7 \cdot 0 + 7 \cdot 0}{8 - 8 lim_{x \to - \infty} \frac{1}{x^{2}} + lim_{x \to - \infty} \frac{5}{x^{3}}}$

Step 7

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

$\frac{- 6 + 7 \cdot 0 + 7 \cdot 0}{8 - 8 \cdot 0 + lim_{x \to - \infty} \frac{5}{x^{3}}}$

Step 8

Move the term $5$ outside of the limit because it is constant with respect to $x$ .

$\frac{- 6 + 7 \cdot 0 + 7 \cdot 0}{8 - 8 \cdot 0 + 5 lim_{x \to - \infty} \frac{1}{x^{3}}}$

Step 9

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

$\frac{- 6 + 7 \cdot 0 + 7 \cdot 0}{8 - 8 \cdot 0 + 5 \cdot 0}$

Step 10

Simplify the answer.

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Step 10.1

Simplify the numerator.

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Step 10.1.1

Multiply $7$ by $0$ .

$\frac{- 6 + 0 + 7 \cdot 0}{8 - 8 \cdot 0 + 5 \cdot 0}$

Step 10.1.2

Multiply $7$ by $0$ .

$\frac{- 6 + 0 + 0}{8 - 8 \cdot 0 + 5 \cdot 0}$

Step 10.1.3

Add $- 6 + 0$ and $0$ .

$\frac{- 6 + 0}{8 - 8 \cdot 0 + 5 \cdot 0}$

Step 10.1.4

Add $- 6$ and $0$ .

$\frac{- 6}{8 - 8 \cdot 0 + 5 \cdot 0}$

Step 10.2

Simplify the denominator.

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Step 10.2.1

Multiply $- 8$ by $0$ .

$\frac{- 6}{8 + 0 + 5 \cdot 0}$

Step 10.2.2

Multiply $5$ by $0$ .

$\frac{- 6}{8 + 0 + 0}$

Step 10.2.3

Add $8 + 0$ and $0$ .

$\frac{- 6}{8 + 0}$

Step 10.2.4

Add $8$ and $0$ .

$\frac{- 6}{8}$

Step 10.3

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

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Step 10.3.1

Factor $2$ out of $- 6$ .

$\frac{2 (- 3)}{8}$

Step 10.3.2

Cancel the common factors.

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Step 10.3.2.1

Factor $2$ out of $8$ .

$\frac{2 \cdot - 3}{2 \cdot 4}$

Step 10.3.2.2

Cancel the common factor.

$\frac{2 \cdot - 3}{2 \cdot 4}$

Step 10.3.2.3

Rewrite the expression.

$\frac{- 3}{4}$

Step 10.4

Move the negative in front of the fraction.

$- \frac{3}{4}$

Step 11

The result can be shown in multiple forms.

Exact Form:

$- \frac{3}{4}$

Decimal Form:

$- 0.75$