Calculus Examples

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

Step 1

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

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

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 $x^{4}$ and $x^{3}$ .

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Step 2.1.1.1

Factor $x^{3}$ out of $x^{4}$ .

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

Step 2.1.1.2

Cancel the common factors.

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Step 2.1.1.2.1

Multiply by $1$ .

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

Step 2.1.1.2.2

Cancel the common factor.

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

Step 2.1.1.2.3

Rewrite the expression.

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

Step 2.1.1.2.4

Divide $x$ by $1$ .

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

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 $- 3 x^{2}$ .

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

Step 2.1.2.2

Cancel the common factors.

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Step 2.1.2.2.1

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

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

Step 2.1.2.2.2

Cancel the common factor.

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

Step 2.1.2.2.3

Rewrite the expression.

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

Step 2.1.3

Move the negative in front of the fraction.

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

Step 2.1.4

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

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Step 2.1.4.1

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

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

Step 2.1.4.2

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

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

Step 2.1.4.3

Cancel the common factors.

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Step 2.1.4.3.1

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

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

Step 2.1.4.3.2

Cancel the common factor.

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

Step 2.1.4.3.3

Rewrite the expression.

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

Step 2.2

Simplify each term.

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Step 2.2.1

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

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Step 2.2.1.1

Cancel the common factor.

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

Step 2.2.1.2

Rewrite the expression.

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

Step 2.2.2

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

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Step 2.2.2.1

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

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

Step 2.2.2.2

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

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

Step 2.2.2.3

Cancel the common factors.

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Step 2.2.2.3.1

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

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

Step 2.2.2.3.2

Cancel the common factor.

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

Step 2.2.2.3.3

Rewrite the expression.

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

Step 3

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

$lim_{x \to \infty} \frac{x - 0 + \frac{1}{x^{2}}}{1 - \frac{1}{x^{2}} + \frac{2}{x^{3}}}$

Step 4

As $x$ approaches $\infty$ , the fraction $\frac{1}{x^{2}}$ approaches $0$ .

$lim_{x \to \infty} \frac{x - 0 + 0}{1 - \frac{1}{x^{2}} + \frac{2}{x^{3}}}$

Step 5

As $x$ approaches $\infty$ , the fraction $\frac{1}{x^{2}}$ approaches $0$ .

$lim_{x \to \infty} \frac{x - 0 + 0}{1 - 0 + \frac{2}{x^{3}}}$

Step 6

As $x$ approaches $\infty$ , the fraction $\frac{2}{x^{3}}$ approaches $0$ .

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

Step 7

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

$\infty$