Calculus Examples

$lim_{x \to \infty} \frac{5 x - 28 x^{3}}{5 x^{2} - 14 x^{3} - 9}$

Step 1

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

$lim_{x \to \infty} \frac{\frac{5 x}{x^{3}} + \frac{- 28 x^{3}}{x^{3}}}{\frac{5 x^{2}}{x^{3}} + \frac{- 14 x^{3}}{x^{3}} + \frac{- 9}{x^{3}}}$

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 $x$ and $x^{3}$ .

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Step 2.1.1.1

Factor $x$ out of $5 x$ .

$lim_{x \to \infty} \frac{\frac{x \cdot 5}{x^{3}} + \frac{- 28 x^{3}}{x^{3}}}{\frac{5 x^{2}}{x^{3}} + \frac{- 14 x^{3}}{x^{3}} + \frac{- 9}{x^{3}}}$

Step 2.1.1.2

Cancel the common factors.

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Step 2.1.1.2.1

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

$lim_{x \to \infty} \frac{\frac{x \cdot 5}{x \cdot x^{2}} + \frac{- 28 x^{3}}{x^{3}}}{\frac{5 x^{2}}{x^{3}} + \frac{- 14 x^{3}}{x^{3}} + \frac{- 9}{x^{3}}}$

Step 2.1.1.2.2

Cancel the common factor.

$lim_{x \to \infty} \frac{\frac{x \cdot 5}{x \cdot x^{2}} + \frac{- 28 x^{3}}{x^{3}}}{\frac{5 x^{2}}{x^{3}} + \frac{- 14 x^{3}}{x^{3}} + \frac{- 9}{x^{3}}}$

Step 2.1.1.2.3

Rewrite the expression.

$lim_{x \to \infty} \frac{\frac{5}{x^{2}} + \frac{- 28 x^{3}}{x^{3}}}{\frac{5 x^{2}}{x^{3}} + \frac{- 14 x^{3}}{x^{3}} + \frac{- 9}{x^{3}}}$

Step 2.1.2

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

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Step 2.1.2.1

Cancel the common factor.

$lim_{x \to \infty} \frac{\frac{5}{x^{2}} + \frac{- 28 x^{3}}{x^{3}}}{\frac{5 x^{2}}{x^{3}} + \frac{- 14 x^{3}}{x^{3}} + \frac{- 9}{x^{3}}}$

Step 2.1.2.2

Divide $- 28$ by $1$ .

$lim_{x \to \infty} \frac{\frac{5}{x^{2}} - 28}{\frac{5 x^{2}}{x^{3}} + \frac{- 14 x^{3}}{x^{3}} + \frac{- 9}{x^{3}}}$

Step 2.2

Simplify each term.

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Step 2.2.1

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

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Step 2.2.1.1

Factor $x^{2}$ out of $5 x^{2}$ .

$lim_{x \to \infty} \frac{\frac{5}{x^{2}} - 28}{\frac{x^{2} \cdot 5}{x^{3}} + \frac{- 14 x^{3}}{x^{3}} + \frac{- 9}{x^{3}}}$

Step 2.2.1.2

Cancel the common factors.

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Step 2.2.1.2.1

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

$lim_{x \to \infty} \frac{\frac{5}{x^{2}} - 28}{\frac{x^{2} \cdot 5}{x^{2} x} + \frac{- 14 x^{3}}{x^{3}} + \frac{- 9}{x^{3}}}$

Step 2.2.1.2.2

Cancel the common factor.

$lim_{x \to \infty} \frac{\frac{5}{x^{2}} - 28}{\frac{x^{2} \cdot 5}{x^{2} x} + \frac{- 14 x^{3}}{x^{3}} + \frac{- 9}{x^{3}}}$

Step 2.2.1.2.3

Rewrite the expression.

$lim_{x \to \infty} \frac{\frac{5}{x^{2}} - 28}{\frac{5}{x} + \frac{- 14 x^{3}}{x^{3}} + \frac{- 9}{x^{3}}}$

Step 2.2.2

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

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Step 2.2.2.1

Cancel the common factor.

$lim_{x \to \infty} \frac{\frac{5}{x^{2}} - 28}{\frac{5}{x} + \frac{- 14 x^{3}}{x^{3}} + \frac{- 9}{x^{3}}}$

Step 2.2.2.2

Divide $- 14$ by $1$ .

$lim_{x \to \infty} \frac{\frac{5}{x^{2}} - 28}{\frac{5}{x} - 14 + \frac{- 9}{x^{3}}}$

Step 2.2.3

Move the negative in front of the fraction.

$lim_{x \to \infty} \frac{\frac{5}{x^{2}} - 28}{\frac{5}{x} - 14 - \frac{9}{x^{3}}}$

Step 2.3

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

$\frac{lim_{x \to \infty} \frac{5}{x^{2}} - 28}{lim_{x \to \infty} \frac{5}{x} - 14 - \frac{9}{x^{3}}}$

Step 2.4

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

$\frac{lim_{x \to \infty} \frac{5}{x^{2}} - lim_{x \to \infty} 28}{lim_{x \to \infty} \frac{5}{x} - 14 - \frac{9}{x^{3}}}$

Step 2.5

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

$\frac{5 lim_{x \to \infty} \frac{1}{x^{2}} - lim_{x \to \infty} 28}{lim_{x \to \infty} \frac{5}{x} - 14 - \frac{9}{x^{3}}}$

Step 3

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

$\frac{5 \cdot 0 - lim_{x \to \infty} 28}{lim_{x \to \infty} \frac{5}{x} - 14 - \frac{9}{x^{3}}}$

Step 4

Evaluate the limit.

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Step 4.1

Evaluate the limit of $28$ which is constant as $x$ approaches $\infty$ .

$\frac{5 \cdot 0 - 1 \cdot 28}{lim_{x \to \infty} \frac{5}{x} - 14 - \frac{9}{x^{3}}}$

Step 4.2

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

$\frac{5 \cdot 0 - 1 \cdot 28}{lim_{x \to \infty} \frac{5}{x} - lim_{x \to \infty} 14 - lim_{x \to \infty} \frac{9}{x^{3}}}$

Step 4.3

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

$\frac{5 \cdot 0 - 1 \cdot 28}{5 lim_{x \to \infty} \frac{1}{x} - lim_{x \to \infty} 14 - lim_{x \to \infty} \frac{9}{x^{3}}}$

Step 5

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

$\frac{5 \cdot 0 - 1 \cdot 28}{5 \cdot 0 - lim_{x \to \infty} 14 - lim_{x \to \infty} \frac{9}{x^{3}}}$

Step 6

Evaluate the limit.

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Step 6.1

Evaluate the limit of $14$ which is constant as $x$ approaches $\infty$ .

$\frac{5 \cdot 0 - 1 \cdot 28}{5 \cdot 0 - 1 \cdot 14 - lim_{x \to \infty} \frac{9}{x^{3}}}$

Step 6.2

Move the term $9$ outside of the limit because it is constant with respect to $x$ .

$\frac{5 \cdot 0 - 1 \cdot 28}{5 \cdot 0 - 1 \cdot 14 - 9 lim_{x \to \infty} \frac{1}{x^{3}}}$

Step 7

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

$\frac{5 \cdot 0 - 1 \cdot 28}{5 \cdot 0 - 1 \cdot 14 - 9 \cdot 0}$

Step 8

Simplify the answer.

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Step 8.1

Simplify the numerator.

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Step 8.1.1

Multiply $5$ by $0$ .

$\frac{0 - 1 \cdot 28}{5 \cdot 0 - 1 \cdot 14 - 9 \cdot 0}$

Step 8.1.2

Multiply $- 1$ by $28$ .

$\frac{0 - 28}{5 \cdot 0 - 1 \cdot 14 - 9 \cdot 0}$

Step 8.1.3

Subtract $28$ from $0$ .

$\frac{- 28}{5 \cdot 0 - 1 \cdot 14 - 9 \cdot 0}$

Step 8.2

Simplify the denominator.

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Step 8.2.1

Multiply $5$ by $0$ .

$\frac{- 28}{0 - 1 \cdot 14 - 9 \cdot 0}$

Step 8.2.2

Multiply $- 1$ by $14$ .

$\frac{- 28}{0 - 14 - 9 \cdot 0}$

Step 8.2.3

Multiply $- 9$ by $0$ .

$\frac{- 28}{0 - 14 + 0}$

Step 8.2.4

Add $0 - 14$ and $0$ .

$\frac{- 28}{0 - 14}$

Step 8.2.5

Subtract $14$ from $0$ .

$\frac{- 28}{- 14}$

Step 8.3

Divide $- 28$ by $- 14$ .

$2$