Calculus Examples

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

Step 1

Divide the numerator and denominator by the highest power of $x$ in the denominator.

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

Step 2

Reduce the expression by cancelling the common factors.

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Step 2.1

Cancel the common factor of $x^{6}$ and $x^{3}$ .

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Step 2.1.1

Factor $x^{3}$ out of $x^{6}$ .

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

Step 2.1.2

Cancel the common factors.

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Step 2.1.2.1

Multiply by $1$ .

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

Step 2.1.2.2

Cancel the common factor.

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

Step 2.1.2.3

Rewrite the expression.

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

Step 2.1.2.4

Divide $x^{3}$ by $1$ .

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

Step 2.2

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

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

Step 3

As $x$ approaches $\infty$ , the fraction $\frac{4}{x^{3}}$ approaches $0$ .

$lim_{x \to \infty} \frac{{(\frac{x^{3}}{x^{3}} + 0)}^{2} - x^{3}}{1}$

Step 4

Since its numerator is unbounded while its denominator approaches a constant number, the fraction $\frac{{(\frac{x^{3}}{x^{3}} + 0)}^{2} - x^{3}}{1}$ approaches negative infinity.

$- \infty$