Calculus Examples

$lim_{x \to - \infty} \frac{4 x^{5} + 1}{x + x^{3}}$

Step 1

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

$lim_{x \to - \infty} \frac{\frac{4 x^{5}}{x^{3}} + \frac{1}{x^{3}}}{\frac{x}{x^{3}} + \frac{x^{3}}{x^{3}}}$

Step 2

Simplify terms.

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Step 2.1

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

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Step 2.1.1

Factor $x^{3}$ out of $4 x^{5}$ .

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

Step 2.1.2

Cancel the common factors.

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Step 2.1.2.1

Multiply by $1$ .

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

Step 2.1.2.2

Cancel the common factor.

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

Step 2.1.2.3

Rewrite the expression.

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

Step 2.1.2.4

Divide $4 x^{2}$ by $1$ .

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

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

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

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

Step 2.2.1.2

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

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

Step 2.2.1.3

Cancel the common factors.

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Step 2.2.1.3.1

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

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

Step 2.2.1.3.2

Cancel the common factor.

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

Step 2.2.1.3.3

Rewrite the expression.

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

Step 2.2.2

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

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Step 2.2.2.1

Cancel the common factor.

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

Step 2.2.2.2

Rewrite the expression.

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

Step 3

As $x$ approaches $- \infty$ , the fraction $\frac{1}{x^{3}}$ approaches $0$ .

$lim_{x \to - \infty} \frac{4 x^{2} + 0}{\frac{1}{x^{2}} + 1}$

Step 4

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

$lim_{x \to - \infty} \frac{4 x^{2} + 0}{0 + 1}$

Step 5

Since its numerator is unbounded while its denominator approaches a constant number, the fraction $\frac{4 x^{2} + 0}{0 + 1}$ approaches infinity.

$\infty$