Calculus Examples

$lim_{x \to - \infty} \frac{4 x^{3} + 6 x^{2} - 2}{5 - 4 x + 2 x^{2}}$

Step 1

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

$lim_{x \to - \infty} \frac{\frac{4 x^{3}}{x^{2}} + \frac{6 x^{2}}{x^{2}} + \frac{- 2}{x^{2}}}{\frac{5}{x^{2}} + \frac{- 4 x}{x^{2}} + \frac{2 x^{2}}{x^{2}}}$

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 $x^{3}$ and $x^{2}$ .

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Step 2.1.1.1

Factor $x^{2}$ out of $4 x^{3}$ .

$lim_{x \to - \infty} \frac{\frac{x^{2} (4 x)}{x^{2}} + \frac{6 x^{2}}{x^{2}} + \frac{- 2}{x^{2}}}{\frac{5}{x^{2}} + \frac{- 4 x}{x^{2}} + \frac{2 x^{2}}{x^{2}}}$

Step 2.1.1.2

Cancel the common factors.

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Step 2.1.1.2.1

Multiply by $1$ .

$lim_{x \to - \infty} \frac{\frac{x^{2} (4 x)}{x^{2} \cdot 1} + \frac{6 x^{2}}{x^{2}} + \frac{- 2}{x^{2}}}{\frac{5}{x^{2}} + \frac{- 4 x}{x^{2}} + \frac{2 x^{2}}{x^{2}}}$

Step 2.1.1.2.2

Cancel the common factor.

$lim_{x \to - \infty} \frac{\frac{x^{2} (4 x)}{x^{2} \cdot 1} + \frac{6 x^{2}}{x^{2}} + \frac{- 2}{x^{2}}}{\frac{5}{x^{2}} + \frac{- 4 x}{x^{2}} + \frac{2 x^{2}}{x^{2}}}$

Step 2.1.1.2.3

Rewrite the expression.

$lim_{x \to - \infty} \frac{\frac{4 x}{1} + \frac{6 x^{2}}{x^{2}} + \frac{- 2}{x^{2}}}{\frac{5}{x^{2}} + \frac{- 4 x}{x^{2}} + \frac{2 x^{2}}{x^{2}}}$

Step 2.1.1.2.4

Divide $4 x$ by $1$ .

$lim_{x \to - \infty} \frac{4 x + \frac{6 x^{2}}{x^{2}} + \frac{- 2}{x^{2}}}{\frac{5}{x^{2}} + \frac{- 4 x}{x^{2}} + \frac{2 x^{2}}{x^{2}}}$

Step 2.1.2

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

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Step 2.1.2.1

Cancel the common factor.

$lim_{x \to - \infty} \frac{4 x + \frac{6 x^{2}}{x^{2}} + \frac{- 2}{x^{2}}}{\frac{5}{x^{2}} + \frac{- 4 x}{x^{2}} + \frac{2 x^{2}}{x^{2}}}$

Step 2.1.2.2

Divide $6$ by $1$ .

$lim_{x \to - \infty} \frac{4 x + 6 + \frac{- 2}{x^{2}}}{\frac{5}{x^{2}} + \frac{- 4 x}{x^{2}} + \frac{2 x^{2}}{x^{2}}}$

Step 2.1.3

Move the negative in front of the fraction.

$lim_{x \to - \infty} \frac{4 x + 6 - \frac{2}{x^{2}}}{\frac{5}{x^{2}} + \frac{- 4 x}{x^{2}} + \frac{2 x^{2}}{x^{2}}}$

Step 2.2

Simplify each term.

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Step 2.2.1

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

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Step 2.2.1.1

Factor $x$ out of $- 4 x$ .

$lim_{x \to - \infty} \frac{4 x + 6 - \frac{2}{x^{2}}}{\frac{5}{x^{2}} + \frac{x \cdot - 4}{x^{2}} + \frac{2 x^{2}}{x^{2}}}$

Step 2.2.1.2

Cancel the common factors.

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Step 2.2.1.2.1

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

$lim_{x \to - \infty} \frac{4 x + 6 - \frac{2}{x^{2}}}{\frac{5}{x^{2}} + \frac{x \cdot - 4}{x \cdot x} + \frac{2 x^{2}}{x^{2}}}$

Step 2.2.1.2.2

Cancel the common factor.

$lim_{x \to - \infty} \frac{4 x + 6 - \frac{2}{x^{2}}}{\frac{5}{x^{2}} + \frac{x \cdot - 4}{x \cdot x} + \frac{2 x^{2}}{x^{2}}}$

Step 2.2.1.2.3

Rewrite the expression.

$lim_{x \to - \infty} \frac{4 x + 6 - \frac{2}{x^{2}}}{\frac{5}{x^{2}} + \frac{- 4}{x} + \frac{2 x^{2}}{x^{2}}}$

Step 2.2.2

Move the negative in front of the fraction.

$lim_{x \to - \infty} \frac{4 x + 6 - \frac{2}{x^{2}}}{\frac{5}{x^{2}} - \frac{4}{x} + \frac{2 x^{2}}{x^{2}}}$

Step 2.2.3

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

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Step 2.2.3.1

Cancel the common factor.

$lim_{x \to - \infty} \frac{4 x + 6 - \frac{2}{x^{2}}}{\frac{5}{x^{2}} - \frac{4}{x} + \frac{2 x^{2}}{x^{2}}}$

Step 2.2.3.2

Divide $2$ by $1$ .

$lim_{x \to - \infty} \frac{4 x + 6 - \frac{2}{x^{2}}}{\frac{5}{x^{2}} - \frac{4}{x} + 2}$

Step 3

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

$lim_{x \to - \infty} \frac{4 x + 6 - 0}{\frac{5}{x^{2}} - \frac{4}{x} + 2}$

Step 4

As $x$ approaches $- \infty$ , the fraction $\frac{5}{x^{2}}$ approaches $0$ .

$lim_{x \to - \infty} \frac{4 x + 6 - 0}{0 - \frac{4}{x} + 2}$

Step 5

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

$lim_{x \to - \infty} \frac{4 x + 6 - 0}{0 - 0 + 2}$

Step 6

Since its numerator is unbounded while its denominator approaches a constant number, the fraction $\frac{4 x + 6 - 0}{0 - 0 + 2}$ approaches negative infinity.

$- \infty$