Calculus Examples

$lim_{x \to \infty} \frac{6 x^{5} - 12 x^{2} + 14 x}{3 x^{5} + 13 x^{3}}$

Step 1

Divide the numerator and denominator by the highest power of $x$ in the denominator, which is $x^{5}$ .

$lim_{x \to \infty} \frac{\frac{6 x^{5}}{x^{5}} + \frac{- 12 x^{2}}{x^{5}} + \frac{14 x}{x^{5}}}{\frac{3 x^{5}}{x^{5}} + \frac{13 x^{3}}{x^{5}}}$

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 $x^{5}$ .

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Step 2.1.1.1

Cancel the common factor.

$lim_{x \to \infty} \frac{\frac{6 x^{5}}{x^{5}} + \frac{- 12 x^{2}}{x^{5}} + \frac{14 x}{x^{5}}}{\frac{3 x^{5}}{x^{5}} + \frac{13 x^{3}}{x^{5}}}$

Step 2.1.1.2

Divide $6$ by $1$ .

$lim_{x \to \infty} \frac{6 + \frac{- 12 x^{2}}{x^{5}} + \frac{14 x}{x^{5}}}{\frac{3 x^{5}}{x^{5}} + \frac{13 x^{3}}{x^{5}}}$

Step 2.1.2

Cancel the common factor of $x^{2}$ and $x^{5}$ .

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Step 2.1.2.1

Factor $x^{2}$ out of $- 12 x^{2}$ .

$lim_{x \to \infty} \frac{6 + \frac{x^{2} \cdot - 12}{x^{5}} + \frac{14 x}{x^{5}}}{\frac{3 x^{5}}{x^{5}} + \frac{13 x^{3}}{x^{5}}}$

Step 2.1.2.2

Cancel the common factors.

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Step 2.1.2.2.1

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

$lim_{x \to \infty} \frac{6 + \frac{x^{2} \cdot - 12}{x^{2} x^{3}} + \frac{14 x}{x^{5}}}{\frac{3 x^{5}}{x^{5}} + \frac{13 x^{3}}{x^{5}}}$

Step 2.1.2.2.2

Cancel the common factor.

$lim_{x \to \infty} \frac{6 + \frac{x^{2} \cdot - 12}{x^{2} x^{3}} + \frac{14 x}{x^{5}}}{\frac{3 x^{5}}{x^{5}} + \frac{13 x^{3}}{x^{5}}}$

Step 2.1.2.2.3

Rewrite the expression.

$lim_{x \to \infty} \frac{6 + \frac{- 12}{x^{3}} + \frac{14 x}{x^{5}}}{\frac{3 x^{5}}{x^{5}} + \frac{13 x^{3}}{x^{5}}}$

Step 2.1.3

Move the negative in front of the fraction.

$lim_{x \to \infty} \frac{6 - \frac{12}{x^{3}} + \frac{14 x}{x^{5}}}{\frac{3 x^{5}}{x^{5}} + \frac{13 x^{3}}{x^{5}}}$

Step 2.1.4

Cancel the common factor of $x$ and $x^{5}$ .

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Step 2.1.4.1

Factor $x$ out of $14 x$ .

$lim_{x \to \infty} \frac{6 - \frac{12}{x^{3}} + \frac{x \cdot 14}{x^{5}}}{\frac{3 x^{5}}{x^{5}} + \frac{13 x^{3}}{x^{5}}}$

Step 2.1.4.2

Cancel the common factors.

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Step 2.1.4.2.1

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

$lim_{x \to \infty} \frac{6 - \frac{12}{x^{3}} + \frac{x \cdot 14}{x \cdot x^{4}}}{\frac{3 x^{5}}{x^{5}} + \frac{13 x^{3}}{x^{5}}}$

Step 2.1.4.2.2

Cancel the common factor.

$lim_{x \to \infty} \frac{6 - \frac{12}{x^{3}} + \frac{x \cdot 14}{x \cdot x^{4}}}{\frac{3 x^{5}}{x^{5}} + \frac{13 x^{3}}{x^{5}}}$

Step 2.1.4.2.3

Rewrite the expression.

$lim_{x \to \infty} \frac{6 - \frac{12}{x^{3}} + \frac{14}{x^{4}}}{\frac{3 x^{5}}{x^{5}} + \frac{13 x^{3}}{x^{5}}}$

Step 2.2

Simplify each term.

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Step 2.2.1

Cancel the common factor of $x^{5}$ .

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Step 2.2.1.1

Cancel the common factor.

$lim_{x \to \infty} \frac{6 - \frac{12}{x^{3}} + \frac{14}{x^{4}}}{\frac{3 x^{5}}{x^{5}} + \frac{13 x^{3}}{x^{5}}}$

Step 2.2.1.2

Divide $3$ by $1$ .

$lim_{x \to \infty} \frac{6 - \frac{12}{x^{3}} + \frac{14}{x^{4}}}{3 + \frac{13 x^{3}}{x^{5}}}$

Step 2.2.2

Cancel the common factor of $x^{3}$ and $x^{5}$ .

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Step 2.2.2.1

Factor $x^{3}$ out of $13 x^{3}$ .

$lim_{x \to \infty} \frac{6 - \frac{12}{x^{3}} + \frac{14}{x^{4}}}{3 + \frac{x^{3} \cdot 13}{x^{5}}}$

Step 2.2.2.2

Cancel the common factors.

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Step 2.2.2.2.1

Factor $x^{3}$ out of $x^{5}$ .

$lim_{x \to \infty} \frac{6 - \frac{12}{x^{3}} + \frac{14}{x^{4}}}{3 + \frac{x^{3} \cdot 13}{x^{3} x^{2}}}$

Step 2.2.2.2.2

Cancel the common factor.

$lim_{x \to \infty} \frac{6 - \frac{12}{x^{3}} + \frac{14}{x^{4}}}{3 + \frac{x^{3} \cdot 13}{x^{3} x^{2}}}$

Step 2.2.2.2.3

Rewrite the expression.

$lim_{x \to \infty} \frac{6 - \frac{12}{x^{3}} + \frac{14}{x^{4}}}{3 + \frac{13}{x^{2}}}$

Step 2.3

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

$\frac{lim_{x \to \infty} 6 - \frac{12}{x^{3}} + \frac{14}{x^{4}}}{lim_{x \to \infty} 3 + \frac{13}{x^{2}}}$

Step 2.4

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

$\frac{lim_{x \to \infty} 6 - lim_{x \to \infty} \frac{12}{x^{3}} + lim_{x \to \infty} \frac{14}{x^{4}}}{lim_{x \to \infty} 3 + \frac{13}{x^{2}}}$

Step 2.5

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

$\frac{6 - lim_{x \to \infty} \frac{12}{x^{3}} + lim_{x \to \infty} \frac{14}{x^{4}}}{lim_{x \to \infty} 3 + \frac{13}{x^{2}}}$

Step 2.6

Move the term $12$ outside of the limit because it is constant with respect to $x$ .

$\frac{6 - 12 lim_{x \to \infty} \frac{1}{x^{3}} + lim_{x \to \infty} \frac{14}{x^{4}}}{lim_{x \to \infty} 3 + \frac{13}{x^{2}}}$

Step 3

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

$\frac{6 - 12 \cdot 0 + lim_{x \to \infty} \frac{14}{x^{4}}}{lim_{x \to \infty} 3 + \frac{13}{x^{2}}}$

Step 4

Move the term $14$ outside of the limit because it is constant with respect to $x$ .

$\frac{6 - 12 \cdot 0 + 14 lim_{x \to \infty} \frac{1}{x^{4}}}{lim_{x \to \infty} 3 + \frac{13}{x^{2}}}$

Step 5

Since its numerator approaches a real number while its denominator is unbounded, the fraction $\frac{1}{x^{4}}$ approaches $0$ .

$\frac{6 - 12 \cdot 0 + 14 \cdot 0}{lim_{x \to \infty} 3 + \frac{13}{x^{2}}}$

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 $\infty$ .

$\frac{6 - 12 \cdot 0 + 14 \cdot 0}{lim_{x \to \infty} 3 + lim_{x \to \infty} \frac{13}{x^{2}}}$

Step 6.2

Evaluate the limit of $3$ which is constant as $x$ approaches $\infty$ .

$\frac{6 - 12 \cdot 0 + 14 \cdot 0}{3 + lim_{x \to \infty} \frac{13}{x^{2}}}$

Step 6.3

Move the term $13$ outside of the limit because it is constant with respect to $x$ .

$\frac{6 - 12 \cdot 0 + 14 \cdot 0}{3 + 13 lim_{x \to \infty} \frac{1}{x^{2}}}$

Step 7

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

$\frac{6 - 12 \cdot 0 + 14 \cdot 0}{3 + 13 \cdot 0}$

Step 8

Simplify the answer.

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Step 8.1

Simplify the numerator.

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Step 8.1.1

Multiply $- 12$ by $0$ .

$\frac{6 + 0 + 14 \cdot 0}{3 + 13 \cdot 0}$

Step 8.1.2

Multiply $14$ by $0$ .

$\frac{6 + 0 + 0}{3 + 13 \cdot 0}$

Step 8.1.3

Add $6 + 0$ and $0$ .

$\frac{6 + 0}{3 + 13 \cdot 0}$

Step 8.1.4

Add $6$ and $0$ .

$\frac{6}{3 + 13 \cdot 0}$

Step 8.2

Simplify the denominator.

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Step 8.2.1

Multiply $13$ by $0$ .

$\frac{6}{3 + 0}$

Step 8.2.2

Add $3$ and $0$ .

$\frac{6}{3}$

Step 8.3

Divide $6$ by $3$ .

$2$