Calculus Examples

$lim_{x \to - \infty} \frac{1 - x^{3}}{x^{2} + 7 x}$

Step 1

Divide the numerator and denominator by the highest power of $x$ in the denominator, which is $x^{2}$ .

$lim_{x \to - \infty} \frac{\frac{1}{x^{2}} - \frac{x^{3}}{x^{2}}}{\frac{x^{2}}{x^{2}} + \frac{7 x}{x^{2}}}$

Step 2

Simplify terms.

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Step 2.1

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

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Step 2.1.1

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

$lim_{x \to - \infty} \frac{\frac{1}{x^{2}} - \frac{x^{2} x}{x^{2}}}{\frac{x^{2}}{x^{2}} + \frac{7 x}{x^{2}}}$

Step 2.1.2

Cancel the common factors.

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Step 2.1.2.1

Multiply by $1$ .

$lim_{x \to - \infty} \frac{\frac{1}{x^{2}} - \frac{x^{2} x}{x^{2} \cdot 1}}{\frac{x^{2}}{x^{2}} + \frac{7 x}{x^{2}}}$

Step 2.1.2.2

Cancel the common factor.

$lim_{x \to - \infty} \frac{\frac{1}{x^{2}} - \frac{x^{2} x}{x^{2} \cdot 1}}{\frac{x^{2}}{x^{2}} + \frac{7 x}{x^{2}}}$

Step 2.1.2.3

Rewrite the expression.

$lim_{x \to - \infty} \frac{\frac{1}{x^{2}} - \frac{x}{1}}{\frac{x^{2}}{x^{2}} + \frac{7 x}{x^{2}}}$

Step 2.1.2.4

Divide $x$ by $1$ .

$lim_{x \to - \infty} \frac{\frac{1}{x^{2}} - x}{\frac{x^{2}}{x^{2}} + \frac{7 x}{x^{2}}}$

Step 2.2

Simplify each term.

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Step 2.2.1

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

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Step 2.2.1.1

Cancel the common factor.

$lim_{x \to - \infty} \frac{\frac{1}{x^{2}} - x}{\frac{x^{2}}{x^{2}} + \frac{7 x}{x^{2}}}$

Step 2.2.1.2

Rewrite the expression.

$lim_{x \to - \infty} \frac{\frac{1}{x^{2}} - x}{1 + \frac{7 x}{x^{2}}}$

Step 2.2.2

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

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Step 2.2.2.1

Factor $x$ out of $7 x$ .

$lim_{x \to - \infty} \frac{\frac{1}{x^{2}} - x}{1 + \frac{x \cdot 7}{x^{2}}}$

Step 2.2.2.2

Cancel the common factors.

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Step 2.2.2.2.1

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

$lim_{x \to - \infty} \frac{\frac{1}{x^{2}} - x}{1 + \frac{x \cdot 7}{x \cdot x}}$

Step 2.2.2.2.2

Cancel the common factor.

$lim_{x \to - \infty} \frac{\frac{1}{x^{2}} - x}{1 + \frac{x \cdot 7}{x \cdot x}}$

Step 2.2.2.2.3

Rewrite the expression.

$lim_{x \to - \infty} \frac{\frac{1}{x^{2}} - x}{1 + \frac{7}{x}}$

Step 3

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

$lim_{x \to - \infty} \frac{0 - x}{1 + \frac{7}{x}}$

Step 4

As $x$ approaches $- \infty$ , the fraction $\frac{7}{x}$ approaches $0$ .

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

Step 5

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

$\infty$