Calculus Examples

$lim_{x \to \infty} \frac{5 x^{3} + 27}{20 x^{2} + 10 x + 9}$

Step 1

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

$lim_{x \to \infty} \frac{\frac{5 x^{3}}{x^{2}} + \frac{27}{x^{2}}}{\frac{20 x^{2}}{x^{2}} + \frac{10 x}{x^{2}} + \frac{9}{x^{2}}}$

Step 2

Simplify terms.

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Step 2.1

Cancel the common factor of $x^{3}$ and $x^{2}$ .

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Step 2.1.1

Factor $x^{2}$ out of $5 x^{3}$ .

$lim_{x \to \infty} \frac{\frac{x^{2} (5 x)}{x^{2}} + \frac{27}{x^{2}}}{\frac{20 x^{2}}{x^{2}} + \frac{10 x}{x^{2}} + \frac{9}{x^{2}}}$

Step 2.1.2

Cancel the common factors.

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Step 2.1.2.1

Multiply by $1$ .

$lim_{x \to \infty} \frac{\frac{x^{2} (5 x)}{x^{2} \cdot 1} + \frac{27}{x^{2}}}{\frac{20 x^{2}}{x^{2}} + \frac{10 x}{x^{2}} + \frac{9}{x^{2}}}$

Step 2.1.2.2

Cancel the common factor.

$lim_{x \to \infty} \frac{\frac{x^{2} (5 x)}{x^{2} \cdot 1} + \frac{27}{x^{2}}}{\frac{20 x^{2}}{x^{2}} + \frac{10 x}{x^{2}} + \frac{9}{x^{2}}}$

Step 2.1.2.3

Rewrite the expression.

$lim_{x \to \infty} \frac{\frac{5 x}{1} + \frac{27}{x^{2}}}{\frac{20 x^{2}}{x^{2}} + \frac{10 x}{x^{2}} + \frac{9}{x^{2}}}$

Step 2.1.2.4

Divide $5 x$ by $1$ .

$lim_{x \to \infty} \frac{5 x + \frac{27}{x^{2}}}{\frac{20 x^{2}}{x^{2}} + \frac{10 x}{x^{2}} + \frac{9}{x^{2}}}$

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.

$lim_{x \to \infty} \frac{5 x + \frac{27}{x^{2}}}{\frac{20 x^{2}}{x^{2}} + \frac{10 x}{x^{2}} + \frac{9}{x^{2}}}$

Step 2.2.1.2

Divide $20$ by $1$ .

$lim_{x \to \infty} \frac{5 x + \frac{27}{x^{2}}}{20 + \frac{10 x}{x^{2}} + \frac{9}{x^{2}}}$

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 $10 x$ .

$lim_{x \to \infty} \frac{5 x + \frac{27}{x^{2}}}{20 + \frac{x \cdot 10}{x^{2}} + \frac{9}{x^{2}}}$

Step 2.2.2.2

Cancel the common factors.

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Step 2.2.2.2.1

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

$lim_{x \to \infty} \frac{5 x + \frac{27}{x^{2}}}{20 + \frac{x \cdot 10}{x \cdot x} + \frac{9}{x^{2}}}$

Step 2.2.2.2.2

Cancel the common factor.

$lim_{x \to \infty} \frac{5 x + \frac{27}{x^{2}}}{20 + \frac{x \cdot 10}{x \cdot x} + \frac{9}{x^{2}}}$

Step 2.2.2.2.3

Rewrite the expression.

$lim_{x \to \infty} \frac{5 x + \frac{27}{x^{2}}}{20 + \frac{10}{x} + \frac{9}{x^{2}}}$

Step 3

As $x$ approaches $\infty$ , the fraction $\frac{27}{x^{2}}$ approaches $0$ .

$lim_{x \to \infty} \frac{5 x + 0}{20 + \frac{10}{x} + \frac{9}{x^{2}}}$

Step 4

As $x$ approaches $\infty$ , the fraction $\frac{10}{x}$ approaches $0$ .

$lim_{x \to \infty} \frac{5 x + 0}{20 + 0 + \frac{9}{x^{2}}}$

Step 5

As $x$ approaches $\infty$ , the fraction $\frac{9}{x^{2}}$ approaches $0$ .

$lim_{x \to \infty} \frac{5 x + 0}{20 + 0 + 0}$

Step 6

Since its numerator is unbounded while its denominator approaches a constant number, the fraction $\frac{5 x + 0}{20 + 0 + 0}$ approaches infinity.

$\infty$