Calculus Examples

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

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

Step 1

Divide the numerator and denominator by the highest power of

x

$x$ 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}}}

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

Step 2

Reduce the expression by cancelling the common factors.

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Step 2.1

Cancel the common factor of

x^{6}

$x^{6}$ and

x^{3}

$x^{3}$ .

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Step 2.1.1

Factor

x^{3}

$x^{3}$ out of

x^{6}

$x^{6}$ .

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}}}

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

Step 2.1.2

Cancel the common factors.

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Step 2.1.2.1

Multiply by

1

$1$ .

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}}}

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

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

$x$ 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.

$- \infty$