Pre-Algebra Examples

$12 x^{2} + 12 x^{- \frac{1}{2}} + 24 x^{\frac{1}{2}}$

Step 1

Find where the expression $12 x^{2} + \frac{12}{x^{\frac{1}{2}}} + 24 x^{\frac{1}{2}}$ is undefined.

$x \leq 0$

Step 2

Since $12 x^{2} + \frac{12}{x^{\frac{1}{2}}} + 24 x^{\frac{1}{2}}$ $\to$ $\infty$ as $x$ $\to$ $0$ from the left and $12 x^{2} + \frac{12}{x^{\frac{1}{2}}} + 24 x^{\frac{1}{2}}$ $\to$ $\infty$ as $x$ $\to$ $0$ from the right, then $x = 0$ is a vertical asymptote.

$x = 0$

Step 3

Since the limit does not exist, there are no horizontal asymptotes.

No Horizontal Asymptotes

Step 4

Find the oblique asymptote using polynomial division.

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Step 4.1

Combine.

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Step 4.1.1

To write $12 x^{2}$ as a fraction with a common denominator, multiply by $\frac{x^{\frac{1}{2}}}{x^{\frac{1}{2}}}$ .

$\frac{12 x^{2} x^{\frac{1}{2}}}{x^{\frac{1}{2}}} + \frac{12}{x^{\frac{1}{2}}} + 24 x^{\frac{1}{2}}$

Step 4.1.2

Combine the numerators over the common denominator.

$\frac{12 x^{2} x^{\frac{1}{2}} + 12}{x^{\frac{1}{2}}} + 24 x^{\frac{1}{2}}$

Step 4.1.3

Simplify the numerator.

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Step 4.1.3.1

Factor $12$ out of $12 x^{2} x^{\frac{1}{2}} + 12$ .

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Step 4.1.3.1.1

Factor $12$ out of $12 x^{2} x^{\frac{1}{2}}$ .

$\frac{12 (x^{2} x^{\frac{1}{2}}) + 12}{x^{\frac{1}{2}}} + 24 x^{\frac{1}{2}}$

Step 4.1.3.1.2

Factor $12$ out of $12$ .

$\frac{12 (x^{2} x^{\frac{1}{2}}) + 12 (1)}{x^{\frac{1}{2}}} + 24 x^{\frac{1}{2}}$

Step 4.1.3.1.3

Factor $12$ out of $12 (x^{2} x^{\frac{1}{2}}) + 12 (1)$ .

$\frac{12 (x^{2} x^{\frac{1}{2}} + 1)}{x^{\frac{1}{2}}} + 24 x^{\frac{1}{2}}$

Step 4.1.3.2

Multiply $x^{2}$ by $x^{\frac{1}{2}}$ by adding the exponents.

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Step 4.1.3.2.1

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{12 (x^{2 + \frac{1}{2}} + 1)}{x^{\frac{1}{2}}} + 24 x^{\frac{1}{2}}$

Step 4.1.3.2.2

To write $2$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{12 (x^{2 \cdot \frac{2}{2} + \frac{1}{2}} + 1)}{x^{\frac{1}{2}}} + 24 x^{\frac{1}{2}}$

Step 4.1.3.2.3

Combine $2$ and $\frac{2}{2}$ .

$\frac{12 (x^{\frac{2 \cdot 2}{2} + \frac{1}{2}} + 1)}{x^{\frac{1}{2}}} + 24 x^{\frac{1}{2}}$

Step 4.1.3.2.4

Combine the numerators over the common denominator.

$\frac{12 (x^{\frac{2 \cdot 2 + 1}{2}} + 1)}{x^{\frac{1}{2}}} + 24 x^{\frac{1}{2}}$

Step 4.1.3.2.5

Simplify the numerator.

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Step 4.1.3.2.5.1

Multiply $2$ by $2$ .

$\frac{12 (x^{\frac{4 + 1}{2}} + 1)}{x^{\frac{1}{2}}} + 24 x^{\frac{1}{2}}$

Step 4.1.3.2.5.2

Add $4$ and $1$ .

$\frac{12 (x^{\frac{5}{2}} + 1)}{x^{\frac{1}{2}}} + 24 x^{\frac{1}{2}}$

Step 4.1.4

To write $24 x^{\frac{1}{2}}$ as a fraction with a common denominator, multiply by $\frac{x^{\frac{1}{2}}}{x^{\frac{1}{2}}}$ .

$\frac{12 (x^{\frac{5}{2}} + 1)}{x^{\frac{1}{2}}} + \frac{24 x^{\frac{1}{2}} x^{\frac{1}{2}}}{x^{\frac{1}{2}}}$

Step 4.1.5

Combine the numerators over the common denominator.

$\frac{12 (x^{\frac{5}{2}} + 1) + 24 x^{\frac{1}{2}} x^{\frac{1}{2}}}{x^{\frac{1}{2}}}$

Step 4.1.6

Simplify the numerator.

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Step 4.1.6.1

Factor $12$ out of $12 (x^{\frac{5}{2}} + 1) + 24 x^{\frac{1}{2}} x^{\frac{1}{2}}$ .

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Step 4.1.6.1.1

Factor $12$ out of $24 x^{\frac{1}{2}} x^{\frac{1}{2}}$ .

$\frac{12 (x^{\frac{5}{2}} + 1) + 12 (2 x^{\frac{1}{2}} x^{\frac{1}{2}})}{x^{\frac{1}{2}}}$

Step 4.1.6.1.2

Factor $12$ out of $12 (x^{\frac{5}{2}} + 1) + 12 (2 x^{\frac{1}{2}} x^{\frac{1}{2}})$ .

$\frac{12 (x^{\frac{5}{2}} + 1 + 2 x^{\frac{1}{2}} x^{\frac{1}{2}})}{x^{\frac{1}{2}}}$

Step 4.1.6.2

Multiply $x^{\frac{1}{2}}$ by $x^{\frac{1}{2}}$ by adding the exponents.

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Step 4.1.6.2.1

Move $x^{\frac{1}{2}}$ .

$\frac{12 (x^{\frac{5}{2}} + 1 + 2 (x^{\frac{1}{2}} x^{\frac{1}{2}}))}{x^{\frac{1}{2}}}$

Step 4.1.6.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{12 (x^{\frac{5}{2}} + 1 + 2 x^{\frac{1}{2} + \frac{1}{2}})}{x^{\frac{1}{2}}}$

Step 4.1.6.2.3

Combine the numerators over the common denominator.

$\frac{12 (x^{\frac{5}{2}} + 1 + 2 x^{\frac{1 + 1}{2}})}{x^{\frac{1}{2}}}$

Step 4.1.6.2.4

Add $1$ and $1$ .

$\frac{12 (x^{\frac{5}{2}} + 1 + 2 x^{\frac{2}{2}})}{x^{\frac{1}{2}}}$

Step 4.1.6.2.5

Divide $2$ by $2$ .

$\frac{12 (x^{\frac{5}{2}} + 1 + 2 x^{1})}{x^{\frac{1}{2}}}$

Step 4.1.6.3

Simplify $2 x^{1}$ .

$\frac{12 (x^{\frac{5}{2}} + 1 + 2 x)}{x^{\frac{1}{2}}}$

Step 4.1.6.4

Reorder terms.

$\frac{12 (2 x + x^{\frac{5}{2}} + 1)}{x^{\frac{1}{2}}}$

Step 4.1.7

Simplify.

$\frac{24 x + 12 x^{\frac{5}{2}} + 12}{x^{\frac{1}{2}}}$

Step 4.2

Factor $12$ out of $24 x + 12 x^{\frac{5}{2}} + 12$ .

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Step 4.2.1

Factor $12$ out of $24 x$ .

$\frac{12 (2 x) + 12 x^{\frac{5}{2}} + 12}{x^{\frac{1}{2}}}$

Step 4.2.2

Factor $12$ out of $12 x^{\frac{5}{2}}$ .

$\frac{12 (2 x) + 12 (x^{\frac{5}{2}}) + 12}{x^{\frac{1}{2}}}$

Step 4.2.3

Factor $12$ out of $12$ .

$\frac{12 (2 x) + 12 (x^{\frac{5}{2}}) + 12 (1)}{x^{\frac{1}{2}}}$

Step 4.2.4

Factor $12$ out of $12 (2 x) + 12 (x^{\frac{5}{2}})$ .

$\frac{12 (2 x + x^{\frac{5}{2}}) + 12 (1)}{x^{\frac{1}{2}}}$

Step 4.2.5

Factor $12$ out of $12 (2 x + x^{\frac{5}{2}}) + 12 (1)$ .

$\frac{12 (2 x + x^{\frac{5}{2}} + 1)}{x^{\frac{1}{2}}}$

Step 4.3

The oblique asymptote is the polynomial portion of the long division result.

$y = \frac{12 (2 x + x^{\frac{5}{2}} + 1)}{x^{\frac{1}{2}}}$

Step 5

This is the set of all asymptotes.

Vertical Asymptotes: $x = 0$

No Horizontal Asymptotes

Oblique Asymptotes: $y = \frac{12 (2 x + x^{\frac{5}{2}} + 1)}{x^{\frac{1}{2}}}$

Step 6