Calculus Examples

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

Step 1

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

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

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.

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

Step 2.1.1.2

Divide $6$ by $1$ .

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

Step 2.1.2

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

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Step 2.1.2.1

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

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

Step 2.1.2.2

Cancel the common factors.

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Step 2.1.2.2.1

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

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

Step 2.1.2.2.2

Cancel the common factor.

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

Step 2.1.2.2.3

Rewrite the expression.

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

Step 2.1.3

Move the negative in front of the fraction.

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

Step 2.1.4

Move the negative in front of the fraction.

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

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

Factor $x$ out of $5 x$ .

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

Step 2.2.1.2

Cancel the common factors.

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Step 2.2.1.2.1

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

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

Step 2.2.1.2.2

Cancel the common factor.

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

Step 2.2.1.2.3

Rewrite the expression.

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

Step 2.2.2

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

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Step 2.2.2.1

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

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

Step 2.2.2.2

Cancel the common factors.

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Step 2.2.2.2.1

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

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

Step 2.2.2.2.2

Cancel the common factor.

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

Step 2.2.2.2.3

Rewrite the expression.

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

Step 2.2.3

Move the negative in front of the fraction.

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

Step 2.2.4

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

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Step 2.2.4.1

Cancel the common factor.

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

Step 2.2.4.2

Divide $- 12$ by $1$ .

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

Step 2.3

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

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

Step 2.4

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

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

Step 2.5

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

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

Step 2.6

Move the term $2$ outside of the limit because it is constant with respect to $x$ .

$\frac{6 - 2 lim_{x \to \infty} \frac{1}{x} - lim_{x \to \infty} \frac{5}{x^{3}}}{lim_{x \to \infty} \frac{5}{x^{2}} - \frac{3}{x} - 12}$

Step 3

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

$\frac{6 - 2 \cdot 0 - lim_{x \to \infty} \frac{5}{x^{3}}}{lim_{x \to \infty} \frac{5}{x^{2}} - \frac{3}{x} - 12}$

Step 4

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

$\frac{6 - 2 \cdot 0 - 5 lim_{x \to \infty} \frac{1}{x^{3}}}{lim_{x \to \infty} \frac{5}{x^{2}} - \frac{3}{x} - 12}$

Step 5

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

$\frac{6 - 2 \cdot 0 - 5 \cdot 0}{lim_{x \to \infty} \frac{5}{x^{2}} - \frac{3}{x} - 12}$

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 $\infty$ .

$\frac{6 - 2 \cdot 0 - 5 \cdot 0}{lim_{x \to \infty} \frac{5}{x^{2}} - lim_{x \to \infty} \frac{3}{x} - lim_{x \to \infty} 12}$

Step 6.2

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

$\frac{6 - 2 \cdot 0 - 5 \cdot 0}{5 lim_{x \to \infty} \frac{1}{x^{2}} - lim_{x \to \infty} \frac{3}{x} - lim_{x \to \infty} 12}$

Step 7

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

$\frac{6 - 2 \cdot 0 - 5 \cdot 0}{5 \cdot 0 - lim_{x \to \infty} \frac{3}{x} - lim_{x \to \infty} 12}$

Step 8

Move the term $3$ outside of the limit because it is constant with respect to $x$ .

$\frac{6 - 2 \cdot 0 - 5 \cdot 0}{5 \cdot 0 - 3 lim_{x \to \infty} \frac{1}{x} - lim_{x \to \infty} 12}$

Step 9

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

$\frac{6 - 2 \cdot 0 - 5 \cdot 0}{5 \cdot 0 - 3 \cdot 0 - lim_{x \to \infty} 12}$

Step 10

Evaluate the limit.

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Step 10.1

Evaluate the limit of $12$ which is constant as $x$ approaches $\infty$ .

$\frac{6 - 2 \cdot 0 - 5 \cdot 0}{5 \cdot 0 - 3 \cdot 0 - 1 \cdot 12}$

Step 10.2

Simplify the answer.

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Step 10.2.1

Simplify the numerator.

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Step 10.2.1.1

Multiply $- 2$ by $0$ .

$\frac{6 + 0 - 5 \cdot 0}{5 \cdot 0 - 3 \cdot 0 - 1 \cdot 12}$

Step 10.2.1.2

Multiply $- 5$ by $0$ .

$\frac{6 + 0 + 0}{5 \cdot 0 - 3 \cdot 0 - 1 \cdot 12}$

Step 10.2.1.3

Add $6 + 0$ and $0$ .

$\frac{6 + 0}{5 \cdot 0 - 3 \cdot 0 - 1 \cdot 12}$

Step 10.2.1.4

Add $6$ and $0$ .

$\frac{6}{5 \cdot 0 - 3 \cdot 0 - 1 \cdot 12}$

Step 10.2.2

Simplify the denominator.

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Step 10.2.2.1

Multiply $5$ by $0$ .

$\frac{6}{0 - 3 \cdot 0 - 1 \cdot 12}$

Step 10.2.2.2

Multiply $- 3$ by $0$ .

$\frac{6}{0 + 0 - 1 \cdot 12}$

Step 10.2.2.3

Multiply $- 1$ by $12$ .

$\frac{6}{0 + 0 - 12}$

Step 10.2.2.4

Add $0$ and $0$ .

$\frac{6}{0 - 12}$

Step 10.2.2.5

Subtract $12$ from $0$ .

$\frac{6}{- 12}$

Step 10.2.3

Cancel the common factor of $6$ and $- 12$ .

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Step 10.2.3.1

Factor $6$ out of $6$ .

$\frac{6 (1)}{- 12}$

Step 10.2.3.2

Cancel the common factors.

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Step 10.2.3.2.1

Factor $6$ out of $- 12$ .

$\frac{6 \cdot 1}{6 \cdot - 2}$

Step 10.2.3.2.2

Cancel the common factor.

$\frac{6 \cdot 1}{6 \cdot - 2}$

Step 10.2.3.2.3

Rewrite the expression.

$\frac{1}{- 2}$

Step 10.2.4

Move the negative in front of the fraction.

$- \frac{1}{2}$

Step 11

The result can be shown in multiple forms.

Exact Form:

$- \frac{1}{2}$

Decimal Form:

$- 0.5$