Calculus Examples

$lim_{x \to \infty} \frac{10 x^{3} + x^{2} - 1}{2 x^{3} + 7 x - 8}$

Step 1

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

$lim_{x \to \infty} \frac{\frac{10 x^{3}}{x^{3}} + \frac{x^{2}}{x^{3}} - \frac{1}{x^{3}}}{\frac{2 x^{3}}{x^{3}} + \frac{7 x}{x^{3}} + \frac{- 8}{x^{3}}}$

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.

$lim_{x \to \infty} \frac{\frac{10 x^{3}}{x^{3}} + \frac{x^{2}}{x^{3}} - \frac{1}{x^{3}}}{\frac{2 x^{3}}{x^{3}} + \frac{7 x}{x^{3}} + \frac{- 8}{x^{3}}}$

Step 2.1.1.2

Divide $10$ by $1$ .

$lim_{x \to \infty} \frac{10 + \frac{x^{2}}{x^{3}} - \frac{1}{x^{3}}}{\frac{2 x^{3}}{x^{3}} + \frac{7 x}{x^{3}} + \frac{- 8}{x^{3}}}$

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 $1$ .

$lim_{x \to \infty} \frac{10 + \frac{x^{2} \cdot 1}{x^{3}} - \frac{1}{x^{3}}}{\frac{2 x^{3}}{x^{3}} + \frac{7 x}{x^{3}} + \frac{- 8}{x^{3}}}$

Step 2.1.2.2

Cancel the common factors.

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Step 2.1.2.2.1

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

$lim_{x \to \infty} \frac{10 + \frac{x^{2} \cdot 1}{x^{2} x} - \frac{1}{x^{3}}}{\frac{2 x^{3}}{x^{3}} + \frac{7 x}{x^{3}} + \frac{- 8}{x^{3}}}$

Step 2.1.2.2.2

Cancel the common factor.

$lim_{x \to \infty} \frac{10 + \frac{x^{2} \cdot 1}{x^{2} x} - \frac{1}{x^{3}}}{\frac{2 x^{3}}{x^{3}} + \frac{7 x}{x^{3}} + \frac{- 8}{x^{3}}}$

Step 2.1.2.2.3

Rewrite the expression.

$lim_{x \to \infty} \frac{10 + \frac{1}{x} - \frac{1}{x^{3}}}{\frac{2 x^{3}}{x^{3}} + \frac{7 x}{x^{3}} + \frac{- 8}{x^{3}}}$

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.

$lim_{x \to \infty} \frac{10 + \frac{1}{x} - \frac{1}{x^{3}}}{\frac{2 x^{3}}{x^{3}} + \frac{7 x}{x^{3}} + \frac{- 8}{x^{3}}}$

Step 2.2.1.2

Divide $2$ by $1$ .

$lim_{x \to \infty} \frac{10 + \frac{1}{x} - \frac{1}{x^{3}}}{2 + \frac{7 x}{x^{3}} + \frac{- 8}{x^{3}}}$

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 $7 x$ .

$lim_{x \to \infty} \frac{10 + \frac{1}{x} - \frac{1}{x^{3}}}{2 + \frac{x \cdot 7}{x^{3}} + \frac{- 8}{x^{3}}}$

Step 2.2.2.2

Cancel the common factors.

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Step 2.2.2.2.1

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

$lim_{x \to \infty} \frac{10 + \frac{1}{x} - \frac{1}{x^{3}}}{2 + \frac{x \cdot 7}{x \cdot x^{2}} + \frac{- 8}{x^{3}}}$

Step 2.2.2.2.2

Cancel the common factor.

$lim_{x \to \infty} \frac{10 + \frac{1}{x} - \frac{1}{x^{3}}}{2 + \frac{x \cdot 7}{x \cdot x^{2}} + \frac{- 8}{x^{3}}}$

Step 2.2.2.2.3

Rewrite the expression.

$lim_{x \to \infty} \frac{10 + \frac{1}{x} - \frac{1}{x^{3}}}{2 + \frac{7}{x^{2}} + \frac{- 8}{x^{3}}}$

Step 2.2.3

Move the negative in front of the fraction.

$lim_{x \to \infty} \frac{10 + \frac{1}{x} - \frac{1}{x^{3}}}{2 + \frac{7}{x^{2}} - \frac{8}{x^{3}}}$

Step 2.3

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

$\frac{lim_{x \to \infty} 10 + \frac{1}{x} - \frac{1}{x^{3}}}{lim_{x \to \infty} 2 + \frac{7}{x^{2}} - \frac{8}{x^{3}}}$

Step 2.4

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

$\frac{lim_{x \to \infty} 10 + lim_{x \to \infty} \frac{1}{x} - lim_{x \to \infty} \frac{1}{x^{3}}}{lim_{x \to \infty} 2 + \frac{7}{x^{2}} - \frac{8}{x^{3}}}$

Step 2.5

Evaluate the limit of $10$ which is constant as $x$ approaches $\infty$ .

$\frac{10 + lim_{x \to \infty} \frac{1}{x} - lim_{x \to \infty} \frac{1}{x^{3}}}{lim_{x \to \infty} 2 + \frac{7}{x^{2}} - \frac{8}{x^{3}}}$

Step 3

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

$\frac{10 + 0 - lim_{x \to \infty} \frac{1}{x^{3}}}{lim_{x \to \infty} 2 + \frac{7}{x^{2}} - \frac{8}{x^{3}}}$

Step 4

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

$\frac{10 + 0 - 0}{lim_{x \to \infty} 2 + \frac{7}{x^{2}} - \frac{8}{x^{3}}}$

Step 5

Evaluate the limit.

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Step 5.1

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

$\frac{10 + 0 - 0}{lim_{x \to \infty} 2 + lim_{x \to \infty} \frac{7}{x^{2}} - lim_{x \to \infty} \frac{8}{x^{3}}}$

Step 5.2

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

$\frac{10 + 0 - 0}{2 + lim_{x \to \infty} \frac{7}{x^{2}} - lim_{x \to \infty} \frac{8}{x^{3}}}$

Step 5.3

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

$\frac{10 + 0 - 0}{2 + 7 lim_{x \to \infty} \frac{1}{x^{2}} - lim_{x \to \infty} \frac{8}{x^{3}}}$

Step 6

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

$\frac{10 + 0 - 0}{2 + 7 \cdot 0 - lim_{x \to \infty} \frac{8}{x^{3}}}$

Step 7

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

$\frac{10 + 0 - 0}{2 + 7 \cdot 0 - 8 lim_{x \to \infty} \frac{1}{x^{3}}}$

Step 8

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

$\frac{10 + 0 - 0}{2 + 7 \cdot 0 - 8 \cdot 0}$

Step 9

Simplify the answer.

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Step 9.1

Cancel the common factor of $10 + 0 - 0$ and $2 + 7 \cdot 0 - 8 \cdot 0$ .

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Step 9.1.1

Rewrite $10$ as $- 1 (- 10)$ .

$\frac{- 1 (- 10) + 0 - 0}{2 + 7 \cdot 0 - 8 \cdot 0}$

Step 9.1.2

Rewrite $0$ as $- 1 (0)$ .

$\frac{- 1 \cdot - 10 - 1 \cdot 0 - 0}{2 + 7 \cdot 0 - 8 \cdot 0}$

Step 9.1.3

Factor $- 1$ out of $- 1 \cdot - 10 - 1 \cdot 0$ .

$\frac{- 1 \cdot (- 10 + 0) - 0}{2 + 7 \cdot 0 - 8 \cdot 0}$

Step 9.1.4

Factor $- 1$ out of $- 0$ .

$\frac{- 1 \cdot (- 10 + 0) - (0)}{2 + 7 \cdot 0 - 8 \cdot 0}$

Step 9.1.5

Factor $- 1$ out of $- 1 \cdot (- 10 + 0) - (0)$ .

$\frac{- 1 \cdot (- 10 + 0 + 0)}{2 + 7 \cdot 0 - 8 \cdot 0}$

Step 9.1.6

Reorder terms.

$\frac{- 1 \cdot (- 10 + 0 + 0)}{2 + 0 \cdot 7 - 8 \cdot 0}$

Step 9.1.7

Factor $2$ out of $- 1 \cdot (- 10 + 0 + 0)$ .

$\frac{2 (- 1 \cdot (- 5 + 0 + 0))}{2 + 0 \cdot 7 - 8 \cdot 0}$

Step 9.1.8

Cancel the common factors.

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Step 9.1.8.1

Factor $2$ out of $2$ .

$\frac{2 (- 1 \cdot (- 5 + 0 + 0))}{2 (1) + 0 \cdot 7 - 8 \cdot 0}$

Step 9.1.8.2

Factor $2$ out of $0 \cdot 7$ .

$\frac{2 (- 1 \cdot (- 5 + 0 + 0))}{2 (1) + 2 (0 \cdot 7) - 8 \cdot 0}$

Step 9.1.8.3

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

$\frac{2 (- 1 \cdot (- 5 + 0 + 0))}{2 (1 + 0 \cdot 7) - 8 \cdot 0}$

Step 9.1.8.4

Factor $2$ out of $- 8 \cdot 0$ .

$\frac{2 (- 1 \cdot (- 5 + 0 + 0))}{2 (1 + 0 \cdot 7) + 2 (- 4 \cdot 0)}$

Step 9.1.8.5

Factor $2$ out of $2 (1 + 0 \cdot 7) + 2 (- 4 \cdot 0)$ .

$\frac{2 (- 1 \cdot (- 5 + 0 + 0))}{2 (1 + 0 \cdot 7 - 4 \cdot 0)}$

Step 9.1.8.6

Cancel the common factor.

$\frac{2 (- 1 \cdot (- 5 + 0 + 0))}{2 (1 + 0 \cdot 7 - 4 \cdot 0)}$

Step 9.1.8.7

Rewrite the expression.

$\frac{- 1 \cdot (- 5 + 0 + 0)}{1 + 0 \cdot 7 - 4 \cdot 0}$

Step 9.2

Simplify the numerator.

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Step 9.2.1

Add $- 5 + 0$ and $0$ .

$\frac{- 1 (- 5 + 0)}{1 + 0 \cdot 7 - 4 \cdot 0}$

Step 9.2.2

Add $- 5$ and $0$ .

$\frac{- 1 \cdot - 5}{1 + 0 \cdot 7 - 4 \cdot 0}$

Step 9.3

Simplify the denominator.

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Step 9.3.1

Multiply $0$ by $7$ .

$\frac{- 1 \cdot - 5}{1 + 0 - 4 \cdot 0}$

Step 9.3.2

Multiply $- 4$ by $0$ .

$\frac{- 1 \cdot - 5}{1 + 0 + 0}$

Step 9.3.3

Add $1 + 0$ and $0$ .

$\frac{- 1 \cdot - 5}{1 + 0}$

Step 9.3.4

Add $1$ and $0$ .

$\frac{- 1 \cdot - 5}{1}$

Step 9.4

Multiply $- 1$ by $- 5$ .

$\frac{5}{1}$

Step 9.5

Divide $5$ by $1$ .

$5$