Calculus Examples

$lim_{x \to \infty} \frac{2 x^{5} + 3 x^{4} - 31 x}{8 x^{4} - 31 x^{2} + 12}$

Step 1

Divide the numerator and denominator by the highest power of $x$ in the denominator, which is $x^{4}$ .

$lim_{x \to \infty} \frac{\frac{2 x^{5}}{x^{4}} + \frac{3 x^{4}}{x^{4}} + \frac{- 31 x}{x^{4}}}{\frac{8 x^{4}}{x^{4}} + \frac{- 31 x^{2}}{x^{4}} + \frac{12}{x^{4}}}$

Step 2

Simplify terms.

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Step 2.1

Simplify each term.

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Step 2.1.1

Cancel the common factor of $x^{5}$ and $x^{4}$ .

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Step 2.1.1.1

Factor $x^{4}$ out of $2 x^{5}$ .

$lim_{x \to \infty} \frac{\frac{x^{4} (2 x)}{x^{4}} + \frac{3 x^{4}}{x^{4}} + \frac{- 31 x}{x^{4}}}{\frac{8 x^{4}}{x^{4}} + \frac{- 31 x^{2}}{x^{4}} + \frac{12}{x^{4}}}$

Step 2.1.1.2

Cancel the common factors.

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Step 2.1.1.2.1

Multiply by $1$ .

$lim_{x \to \infty} \frac{\frac{x^{4} (2 x)}{x^{4} \cdot 1} + \frac{3 x^{4}}{x^{4}} + \frac{- 31 x}{x^{4}}}{\frac{8 x^{4}}{x^{4}} + \frac{- 31 x^{2}}{x^{4}} + \frac{12}{x^{4}}}$

Step 2.1.1.2.2

Cancel the common factor.

$lim_{x \to \infty} \frac{\frac{x^{4} (2 x)}{x^{4} \cdot 1} + \frac{3 x^{4}}{x^{4}} + \frac{- 31 x}{x^{4}}}{\frac{8 x^{4}}{x^{4}} + \frac{- 31 x^{2}}{x^{4}} + \frac{12}{x^{4}}}$

Step 2.1.1.2.3

Rewrite the expression.

$lim_{x \to \infty} \frac{\frac{2 x}{1} + \frac{3 x^{4}}{x^{4}} + \frac{- 31 x}{x^{4}}}{\frac{8 x^{4}}{x^{4}} + \frac{- 31 x^{2}}{x^{4}} + \frac{12}{x^{4}}}$

Step 2.1.1.2.4

Divide $2 x$ by $1$ .

$lim_{x \to \infty} \frac{2 x + \frac{3 x^{4}}{x^{4}} + \frac{- 31 x}{x^{4}}}{\frac{8 x^{4}}{x^{4}} + \frac{- 31 x^{2}}{x^{4}} + \frac{12}{x^{4}}}$

Step 2.1.2

Cancel the common factor of $x^{4}$ .

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Step 2.1.2.1

Cancel the common factor.

$lim_{x \to \infty} \frac{2 x + \frac{3 x^{4}}{x^{4}} + \frac{- 31 x}{x^{4}}}{\frac{8 x^{4}}{x^{4}} + \frac{- 31 x^{2}}{x^{4}} + \frac{12}{x^{4}}}$

Step 2.1.2.2

Divide $3$ by $1$ .

$lim_{x \to \infty} \frac{2 x + 3 + \frac{- 31 x}{x^{4}}}{\frac{8 x^{4}}{x^{4}} + \frac{- 31 x^{2}}{x^{4}} + \frac{12}{x^{4}}}$

Step 2.1.3

Cancel the common factor of $x$ and $x^{4}$ .

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Step 2.1.3.1

Factor $x$ out of $- 31 x$ .

$lim_{x \to \infty} \frac{2 x + 3 + \frac{x \cdot - 31}{x^{4}}}{\frac{8 x^{4}}{x^{4}} + \frac{- 31 x^{2}}{x^{4}} + \frac{12}{x^{4}}}$

Step 2.1.3.2

Cancel the common factors.

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Step 2.1.3.2.1

Factor $x$ out of $x^{4}$ .

$lim_{x \to \infty} \frac{2 x + 3 + \frac{x \cdot - 31}{x \cdot x^{3}}}{\frac{8 x^{4}}{x^{4}} + \frac{- 31 x^{2}}{x^{4}} + \frac{12}{x^{4}}}$

Step 2.1.3.2.2

Cancel the common factor.

$lim_{x \to \infty} \frac{2 x + 3 + \frac{x \cdot - 31}{x \cdot x^{3}}}{\frac{8 x^{4}}{x^{4}} + \frac{- 31 x^{2}}{x^{4}} + \frac{12}{x^{4}}}$

Step 2.1.3.2.3

Rewrite the expression.

$lim_{x \to \infty} \frac{2 x + 3 + \frac{- 31}{x^{3}}}{\frac{8 x^{4}}{x^{4}} + \frac{- 31 x^{2}}{x^{4}} + \frac{12}{x^{4}}}$

Step 2.1.4

Move the negative in front of the fraction.

$lim_{x \to \infty} \frac{2 x + 3 - \frac{31}{x^{3}}}{\frac{8 x^{4}}{x^{4}} + \frac{- 31 x^{2}}{x^{4}} + \frac{12}{x^{4}}}$

Step 2.2

Simplify each term.

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Step 2.2.1

Cancel the common factor of $x^{4}$ .

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Step 2.2.1.1

Cancel the common factor.

$lim_{x \to \infty} \frac{2 x + 3 - \frac{31}{x^{3}}}{\frac{8 x^{4}}{x^{4}} + \frac{- 31 x^{2}}{x^{4}} + \frac{12}{x^{4}}}$

Step 2.2.1.2

Divide $8$ by $1$ .

$lim_{x \to \infty} \frac{2 x + 3 - \frac{31}{x^{3}}}{8 + \frac{- 31 x^{2}}{x^{4}} + \frac{12}{x^{4}}}$

Step 2.2.2

Cancel the common factor of $x^{2}$ and $x^{4}$ .

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Step 2.2.2.1

Factor $x^{2}$ out of $- 31 x^{2}$ .

$lim_{x \to \infty} \frac{2 x + 3 - \frac{31}{x^{3}}}{8 + \frac{x^{2} \cdot - 31}{x^{4}} + \frac{12}{x^{4}}}$

Step 2.2.2.2

Cancel the common factors.

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Step 2.2.2.2.1

Factor $x^{2}$ out of $x^{4}$ .

$lim_{x \to \infty} \frac{2 x + 3 - \frac{31}{x^{3}}}{8 + \frac{x^{2} \cdot - 31}{x^{2} x^{2}} + \frac{12}{x^{4}}}$

Step 2.2.2.2.2

Cancel the common factor.

$lim_{x \to \infty} \frac{2 x + 3 - \frac{31}{x^{3}}}{8 + \frac{x^{2} \cdot - 31}{x^{2} x^{2}} + \frac{12}{x^{4}}}$

Step 2.2.2.2.3

Rewrite the expression.

$lim_{x \to \infty} \frac{2 x + 3 - \frac{31}{x^{3}}}{8 + \frac{- 31}{x^{2}} + \frac{12}{x^{4}}}$

Step 2.2.3

Move the negative in front of the fraction.

$lim_{x \to \infty} \frac{2 x + 3 - \frac{31}{x^{3}}}{8 - \frac{31}{x^{2}} + \frac{12}{x^{4}}}$

Step 3

As $x$ approaches $\infty$ , the fraction $\frac{31}{x^{3}}$ approaches $0$ .

$lim_{x \to \infty} \frac{2 x + 3 - 0}{8 - \frac{31}{x^{2}} + \frac{12}{x^{4}}}$

Step 4

As $x$ approaches $\infty$ , the fraction $\frac{31}{x^{2}}$ approaches $0$ .

$lim_{x \to \infty} \frac{2 x + 3 - 0}{8 - 0 + \frac{12}{x^{4}}}$

Step 5

As $x$ approaches $\infty$ , the fraction $\frac{12}{x^{4}}$ approaches $0$ .

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

Step 6

Since its numerator is unbounded while its denominator approaches a constant number, the fraction $\frac{2 x + 3 - 0}{8 - 0 + 0}$ approaches infinity.

$\infty$