Calculus Examples

$lim_{x \to - \infty} \frac{2 x^{2} - 7 x^{4}}{9 x^{2} + 5 x - 6}$

Step 1

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

$lim_{x \to - \infty} \frac{\frac{2 x^{2}}{x^{2}} + \frac{- 7 x^{4}}{x^{2}}}{\frac{9 x^{2}}{x^{2}} + \frac{5 x}{x^{2}} + \frac{- 6}{x^{2}}}$

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 $x^{2}$ .

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Step 2.1.1.1

Cancel the common factor.

$lim_{x \to - \infty} \frac{\frac{2 x^{2}}{x^{2}} + \frac{- 7 x^{4}}{x^{2}}}{\frac{9 x^{2}}{x^{2}} + \frac{5 x}{x^{2}} + \frac{- 6}{x^{2}}}$

Step 2.1.1.2

Divide $2$ by $1$ .

$lim_{x \to - \infty} \frac{2 + \frac{- 7 x^{4}}{x^{2}}}{\frac{9 x^{2}}{x^{2}} + \frac{5 x}{x^{2}} + \frac{- 6}{x^{2}}}$

Step 2.1.2

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

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Step 2.1.2.1

Factor $x^{2}$ out of $- 7 x^{4}$ .

$lim_{x \to - \infty} \frac{2 + \frac{x^{2} (- 7 x^{2})}{x^{2}}}{\frac{9 x^{2}}{x^{2}} + \frac{5 x}{x^{2}} + \frac{- 6}{x^{2}}}$

Step 2.1.2.2

Cancel the common factors.

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Step 2.1.2.2.1

Multiply by $1$ .

$lim_{x \to - \infty} \frac{2 + \frac{x^{2} (- 7 x^{2})}{x^{2} \cdot 1}}{\frac{9 x^{2}}{x^{2}} + \frac{5 x}{x^{2}} + \frac{- 6}{x^{2}}}$

Step 2.1.2.2.2

Cancel the common factor.

$lim_{x \to - \infty} \frac{2 + \frac{x^{2} (- 7 x^{2})}{x^{2} \cdot 1}}{\frac{9 x^{2}}{x^{2}} + \frac{5 x}{x^{2}} + \frac{- 6}{x^{2}}}$

Step 2.1.2.2.3

Rewrite the expression.

$lim_{x \to - \infty} \frac{2 + \frac{- 7 x^{2}}{1}}{\frac{9 x^{2}}{x^{2}} + \frac{5 x}{x^{2}} + \frac{- 6}{x^{2}}}$

Step 2.1.2.2.4

Divide $- 7 x^{2}$ by $1$ .

$lim_{x \to - \infty} \frac{2 - 7 x^{2}}{\frac{9 x^{2}}{x^{2}} + \frac{5 x}{x^{2}} + \frac{- 6}{x^{2}}}$

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.

$lim_{x \to - \infty} \frac{2 - 7 x^{2}}{\frac{9 x^{2}}{x^{2}} + \frac{5 x}{x^{2}} + \frac{- 6}{x^{2}}}$

Step 2.2.1.2

Divide $9$ by $1$ .

$lim_{x \to - \infty} \frac{2 - 7 x^{2}}{9 + \frac{5 x}{x^{2}} + \frac{- 6}{x^{2}}}$

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 $5 x$ .

$lim_{x \to - \infty} \frac{2 - 7 x^{2}}{9 + \frac{x \cdot 5}{x^{2}} + \frac{- 6}{x^{2}}}$

Step 2.2.2.2

Cancel the common factors.

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Step 2.2.2.2.1

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

$lim_{x \to - \infty} \frac{2 - 7 x^{2}}{9 + \frac{x \cdot 5}{x \cdot x} + \frac{- 6}{x^{2}}}$

Step 2.2.2.2.2

Cancel the common factor.

$lim_{x \to - \infty} \frac{2 - 7 x^{2}}{9 + \frac{x \cdot 5}{x \cdot x} + \frac{- 6}{x^{2}}}$

Step 2.2.2.2.3

Rewrite the expression.

$lim_{x \to - \infty} \frac{2 - 7 x^{2}}{9 + \frac{5}{x} + \frac{- 6}{x^{2}}}$

Step 2.2.3

Move the negative in front of the fraction.

$lim_{x \to - \infty} \frac{2 - 7 x^{2}}{9 + \frac{5}{x} - \frac{6}{x^{2}}}$

Step 3

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

$lim_{x \to - \infty} \frac{2 - 7 x^{2}}{9 + 0 - \frac{6}{x^{2}}}$

Step 4

As $x$ approaches $- \infty$ , the fraction $\frac{6}{x^{2}}$ approaches $0$ .

$lim_{x \to - \infty} \frac{2 - 7 x^{2}}{9 + 0 - 0}$

Step 5

Since its numerator is unbounded while its denominator approaches a constant number, the fraction $\frac{2 - 7 x^{2}}{9 + 0 - 0}$ approaches negative infinity.

$- \infty$