Calculus Examples

$lim_{x \to \infty} \frac{x^{4}}{3 x^{2} - 7 x}$

Step 1

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

$lim_{x \to \infty} \frac{\frac{x^{4}}{x^{2}}}{\frac{3 x^{2}}{x^{2}} + \frac{- 7 x}{x^{2}}}$

Step 2

Simplify terms.

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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 $x^{4}$ .

$lim_{x \to \infty} \frac{\frac{x^{2} x^{2}}{x^{2}}}{\frac{3 x^{2}}{x^{2}} + \frac{- 7 x}{x^{2}}}$

Step 2.1.2

Cancel the common factors.

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Step 2.1.2.1

Multiply by $1$ .

$lim_{x \to \infty} \frac{\frac{x^{2} x^{2}}{x^{2} \cdot 1}}{\frac{3 x^{2}}{x^{2}} + \frac{- 7 x}{x^{2}}}$

Step 2.1.2.2

Cancel the common factor.

$lim_{x \to \infty} \frac{\frac{x^{2} x^{2}}{x^{2} \cdot 1}}{\frac{3 x^{2}}{x^{2}} + \frac{- 7 x}{x^{2}}}$

Step 2.1.2.3

Rewrite the expression.

$lim_{x \to \infty} \frac{\frac{x^{2}}{1}}{\frac{3 x^{2}}{x^{2}} + \frac{- 7 x}{x^{2}}}$

Step 2.1.2.4

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

$lim_{x \to \infty} \frac{x^{2}}{\frac{3 x^{2}}{x^{2}} + \frac{- 7 x}{x^{2}}}$

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.

$lim_{x \to \infty} \frac{x^{2}}{\frac{3 x^{2}}{x^{2}} + \frac{- 7 x}{x^{2}}}$

Step 2.2.1.2

Divide $3$ by $1$ .

$lim_{x \to \infty} \frac{x^{2}}{3 + \frac{- 7 x}{x^{2}}}$

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 $- 7 x$ .

$lim_{x \to \infty} \frac{x^{2}}{3 + \frac{x \cdot - 7}{x^{2}}}$

Step 2.2.2.2

Cancel the common factors.

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Step 2.2.2.2.1

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

$lim_{x \to \infty} \frac{x^{2}}{3 + \frac{x \cdot - 7}{x \cdot x}}$

Step 2.2.2.2.2

Cancel the common factor.

$lim_{x \to \infty} \frac{x^{2}}{3 + \frac{x \cdot - 7}{x \cdot x}}$

Step 2.2.2.2.3

Rewrite the expression.

$lim_{x \to \infty} \frac{x^{2}}{3 + \frac{- 7}{x}}$

Step 2.2.3

Move the negative in front of the fraction.

$lim_{x \to \infty} \frac{x^{2}}{3 - \frac{7}{x}}$

Step 3

As $x$ approaches $\infty$ , the fraction $\frac{7}{x}$ approaches $0$ .

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

Step 4

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

$\infty$