Calculus Examples

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

Step 1

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

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

Step 2

Reduce the expression by cancelling the common factors.

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Step 2.1

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

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Step 2.1.1

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

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

Step 2.1.2

Cancel the common factors.

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Step 2.1.2.1

Multiply by $1$ .

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

Step 2.1.2.2

Cancel the common factor.

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

Step 2.1.2.3

Rewrite the expression.

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

Step 2.1.2.4

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

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

Step 2.2

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

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Step 2.2.1

Cancel the common factor.

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

Step 2.2.2

Divide $3$ by $1$ .

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

Step 3

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

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

Step 4

Since its numerator is unbounded while its denominator approaches a constant number, the fraction $\frac{- 2 x^{2}}{3 - 0}$ approaches negative infinity.

$- \infty$