Calculus Examples

$lim_{x \to \infty} \frac{5 x^{3} + 1}{x - x^{3}}$

Step 1

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

$lim_{x \to \infty} \frac{\frac{5 x^{3}}{x^{3}} + \frac{1}{x^{3}}}{\frac{x}{x^{3}} - \frac{x^{3}}{x^{3}}}$

Step 2

Evaluate the limit.

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Step 2.1

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

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Step 2.1.1

Cancel the common factor.

$lim_{x \to \infty} \frac{\frac{5 x^{3}}{x^{3}} + \frac{1}{x^{3}}}{\frac{x}{x^{3}} - \frac{x^{3}}{x^{3}}}$

Step 2.1.2

Divide $5$ by $1$ .

$lim_{x \to \infty} \frac{5 + \frac{1}{x^{3}}}{\frac{x}{x^{3}} - \frac{x^{3}}{x^{3}}}$

Step 2.2

Simplify each term.

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Step 2.2.1

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

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Step 2.2.1.1

Raise $x$ to the power of $1$ .

$lim_{x \to \infty} \frac{5 + \frac{1}{x^{3}}}{\frac{x^{1}}{x^{3}} - \frac{x^{3}}{x^{3}}}$

Step 2.2.1.2

Factor $x$ out of $x^{1}$ .

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

Step 2.2.1.3

Cancel the common factors.

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Step 2.2.1.3.1

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

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

Step 2.2.1.3.2

Cancel the common factor.

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

Step 2.2.1.3.3

Rewrite the expression.

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

Step 2.2.2

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

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Step 2.2.2.1

Cancel the common factor.

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

Step 2.2.2.2

Rewrite the expression.

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

Step 2.2.3

Multiply $- 1$ by $1$ .

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

Step 2.3

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

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

Step 2.4

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

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

Step 2.5

Evaluate the limit of $5$ which is constant as $x$ approaches $\infty$ .

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

Step 3

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

$\frac{5 + 0}{lim_{x \to \infty} \frac{1}{x^{2}} - 1}$

Step 4

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

$\frac{5 + 0}{lim_{x \to \infty} \frac{1}{x^{2}} - lim_{x \to \infty} 1}$

Step 5

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

$\frac{5 + 0}{0 - lim_{x \to \infty} 1}$

Step 6

Evaluate the limit.

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Step 6.1

Evaluate the limit of $1$ which is constant as $x$ approaches $\infty$ .

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

Step 6.2

Simplify the answer.

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Step 6.2.1

Add $5$ and $0$ .

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

Step 6.2.2

Simplify the denominator.

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Step 6.2.2.1

Multiply $- 1$ by $1$ .

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

Step 6.2.2.2

Subtract $1$ from $0$ .

$\frac{5}{- 1}$

Step 6.2.3

Divide $5$ by $- 1$ .

$- 5$