Finite Math Examples

$p = \frac{12 z + 30}{2 z} \div \frac{16 z + 40}{4 z}$

Step 1

Find where the expression $\frac{12 x + 30}{2 x} \div \frac{16 x + 40}{4 x}$ is undefined.

$x = - \frac{5}{2}, x = 0$

Step 2

The vertical asymptotes occur at areas of infinite discontinuity.

No Vertical Asymptotes

Step 3

Evaluate $lim_{x \to \infty} \frac{12 x + 30}{2 x} \div \frac{16 x + 40}{4 x}$ to find the horizontal asymptote.

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Step 3.1

Reduce.

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Step 3.1.1

Cancel the common factor of $12 x + 30$ and $2$ .

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Step 3.1.1.1

Factor $2$ out of $12 x$ .

$lim_{x \to \infty} \frac{2 (6 x) + 30}{2 x} \div \frac{16 x + 40}{4 x}$

Step 3.1.1.2

Factor $2$ out of $30$ .

$lim_{x \to \infty} \frac{2 (6 x) + 2 \cdot 15}{2 x} \div \frac{16 x + 40}{4 x}$

Step 3.1.1.3

Factor $2$ out of $2 (6 x) + 2 (15)$ .

$lim_{x \to \infty} \frac{2 (6 x + 15)}{2 x} \div \frac{16 x + 40}{4 x}$

Step 3.1.1.4

Cancel the common factors.

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Step 3.1.1.4.1

Factor $2$ out of $2 x$ .

$lim_{x \to \infty} \frac{2 (6 x + 15)}{2 (x)} \div \frac{16 x + 40}{4 x}$

Step 3.1.1.4.2

Cancel the common factor.

$lim_{x \to \infty} \frac{2 (6 x + 15)}{2 x} \div \frac{16 x + 40}{4 x}$

Step 3.1.1.4.3

Rewrite the expression.

$lim_{x \to \infty} \frac{6 x + 15}{x} \div \frac{16 x + 40}{4 x}$

Step 3.1.2

Cancel the common factor of $16 x + 40$ and $4$ .

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Step 3.1.2.1

Factor $4$ out of $16 x$ .

$lim_{x \to \infty} \frac{6 x + 15}{x} \div \frac{4 (4 x) + 40}{4 x}$

Step 3.1.2.2

Factor $4$ out of $40$ .

$lim_{x \to \infty} \frac{6 x + 15}{x} \div \frac{4 (4 x) + 4 \cdot 10}{4 x}$

Step 3.1.2.3

Factor $4$ out of $4 (4 x) + 4 (10)$ .

$lim_{x \to \infty} \frac{6 x + 15}{x} \div \frac{4 (4 x + 10)}{4 x}$

Step 3.1.2.4

Cancel the common factors.

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Step 3.1.2.4.1

Factor $4$ out of $4 x$ .

$lim_{x \to \infty} \frac{6 x + 15}{x} \div \frac{4 (4 x + 10)}{4 (x)}$

Step 3.1.2.4.2

Cancel the common factor.

$lim_{x \to \infty} \frac{6 x + 15}{x} \div \frac{4 (4 x + 10)}{4 x}$

Step 3.1.2.4.3

Rewrite the expression.

$lim_{x \to \infty} \frac{6 x + 15}{x} \div \frac{4 x + 10}{x}$

Step 3.2

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $\infty$ .

$\frac{lim_{x \to \infty} \frac{6 x + 15}{x}}{lim_{x \to \infty} \frac{4 x + 10}{x}}$

Step 3.3

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

$\frac{lim_{x \to \infty} \frac{\frac{6 x}{x} + \frac{15}{x}}{\frac{x}{x}}}{lim_{x \to \infty} \frac{4 x + 10}{x}}$

Step 3.4

Evaluate the limit.

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Step 3.4.1

Cancel the common factor of $x$ .

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Step 3.4.1.1

Cancel the common factor.

$\frac{lim_{x \to \infty} \frac{\frac{6 x}{x} + \frac{15}{x}}{\frac{x}{x}}}{lim_{x \to \infty} \frac{4 x + 10}{x}}$

Step 3.4.1.2

Divide $6$ by $1$ .

$\frac{lim_{x \to \infty} \frac{6 + \frac{15}{x}}{\frac{x}{x}}}{lim_{x \to \infty} \frac{4 x + 10}{x}}$

Step 3.4.2

Cancel the common factor of $x$ .

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Step 3.4.2.1

Cancel the common factor.

$\frac{lim_{x \to \infty} \frac{6 + \frac{15}{x}}{\frac{x}{x}}}{lim_{x \to \infty} \frac{4 x + 10}{x}}$

Step 3.4.2.2

Rewrite the expression.

$\frac{lim_{x \to \infty} \frac{6 + \frac{15}{x}}{1}}{lim_{x \to \infty} \frac{4 x + 10}{x}}$

Step 3.4.3

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $\infty$ .

$\frac{\frac{lim_{x \to \infty} 6 + \frac{15}{x}}{lim_{x \to \infty} 1}}{lim_{x \to \infty} \frac{4 x + 10}{x}}$

Step 3.4.4

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $\infty$ .

$\frac{\frac{lim_{x \to \infty} 6 + lim_{x \to \infty} \frac{15}{x}}{lim_{x \to \infty} 1}}{lim_{x \to \infty} \frac{4 x + 10}{x}}$

Step 3.4.5

Evaluate the limit of $6$ which is constant as $x$ approaches $\infty$ .

$\frac{\frac{6 + lim_{x \to \infty} \frac{15}{x}}{lim_{x \to \infty} 1}}{lim_{x \to \infty} \frac{4 x + 10}{x}}$

Step 3.4.6

Move the term $15$ outside of the limit because it is constant with respect to $x$ .

$\frac{\frac{6 + 15 lim_{x \to \infty} \frac{1}{x}}{lim_{x \to \infty} 1}}{lim_{x \to \infty} \frac{4 x + 10}{x}}$

Step 3.5

Since its numerator approaches a real number while its denominator is unbounded, the fraction $\frac{1}{x}$ approaches $0$ .

$\frac{\frac{6 + 15 \cdot 0}{lim_{x \to \infty} 1}}{lim_{x \to \infty} \frac{4 x + 10}{x}}$

Step 3.6

Evaluate the limit of $1$ which is constant as $x$ approaches $\infty$ .

$\frac{\frac{6 + 15 \cdot 0}{1}}{lim_{x \to \infty} \frac{4 x + 10}{x}}$

Step 3.7

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

$\frac{\frac{6 + 15 \cdot 0}{1}}{lim_{x \to \infty} \frac{\frac{4 x}{x} + \frac{10}{x}}{\frac{x}{x}}}$

Step 3.8

Evaluate the limit.

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Step 3.8.1

Cancel the common factor of $x$ .

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Step 3.8.1.1

Cancel the common factor.

$\frac{\frac{6 + 15 \cdot 0}{1}}{lim_{x \to \infty} \frac{\frac{4 x}{x} + \frac{10}{x}}{\frac{x}{x}}}$

Step 3.8.1.2

Divide $4$ by $1$ .

$\frac{\frac{6 + 15 \cdot 0}{1}}{lim_{x \to \infty} \frac{4 + \frac{10}{x}}{\frac{x}{x}}}$

Step 3.8.2

Cancel the common factor of $x$ .

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Step 3.8.2.1

Cancel the common factor.

$\frac{\frac{6 + 15 \cdot 0}{1}}{lim_{x \to \infty} \frac{4 + \frac{10}{x}}{\frac{x}{x}}}$

Step 3.8.2.2

Rewrite the expression.

$\frac{\frac{6 + 15 \cdot 0}{1}}{lim_{x \to \infty} \frac{4 + \frac{10}{x}}{1}}$

Step 3.8.3

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $\infty$ .

$\frac{\frac{6 + 15 \cdot 0}{1}}{\frac{lim_{x \to \infty} 4 + \frac{10}{x}}{lim_{x \to \infty} 1}}$

Step 3.8.4

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $\infty$ .

$\frac{\frac{6 + 15 \cdot 0}{1}}{\frac{lim_{x \to \infty} 4 + lim_{x \to \infty} \frac{10}{x}}{lim_{x \to \infty} 1}}$

Step 3.8.5

Evaluate the limit of $4$ which is constant as $x$ approaches $\infty$ .

$\frac{\frac{6 + 15 \cdot 0}{1}}{\frac{4 + lim_{x \to \infty} \frac{10}{x}}{lim_{x \to \infty} 1}}$

Step 3.8.6

Move the term $10$ outside of the limit because it is constant with respect to $x$ .

$\frac{\frac{6 + 15 \cdot 0}{1}}{\frac{4 + 10 lim_{x \to \infty} \frac{1}{x}}{lim_{x \to \infty} 1}}$

Step 3.9

Since its numerator approaches a real number while its denominator is unbounded, the fraction $\frac{1}{x}$ approaches $0$ .

$\frac{\frac{6 + 15 \cdot 0}{1}}{\frac{4 + 10 \cdot 0}{lim_{x \to \infty} 1}}$

Step 3.10

Evaluate the limit.

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Step 3.10.1

Evaluate the limit of $1$ which is constant as $x$ approaches $\infty$ .

$\frac{\frac{6 + 15 \cdot 0}{1}}{\frac{4 + 10 \cdot 0}{1}}$

Step 3.10.2

Simplify the answer.

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Step 3.10.2.1

Divide $6 + 15 \cdot 0$ by $1$ .

$\frac{6 + 15 \cdot 0}{\frac{4 + 10 \cdot 0}{1}}$

Step 3.10.2.2

Divide $4 + 10 \cdot 0$ by $1$ .

$\frac{6 + 15 \cdot 0}{4 + 10 \cdot 0}$

Step 3.10.2.3

Cancel the common factor of $6 + 15 \cdot 0$ and $4 + 10 \cdot 0$ .

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Step 3.10.2.3.1

Reorder terms.

$\frac{6 + 0 \cdot 15}{4 + 10 \cdot 0}$

Step 3.10.2.3.2

Factor $2$ out of $6$ .

$\frac{2 (3) + 0 \cdot 15}{4 + 10 \cdot 0}$

Step 3.10.2.3.3

Factor $2$ out of $0 \cdot 15$ .

$\frac{2 (3) + 2 (0 \cdot 15)}{4 + 10 \cdot 0}$

Step 3.10.2.3.4

Factor $2$ out of $2 (3) + 2 (0 \cdot 15)$ .

$\frac{2 (3 + 0 \cdot 15)}{4 + 10 \cdot 0}$

Step 3.10.2.3.5

Cancel the common factors.

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Step 3.10.2.3.5.1

Factor $2$ out of $4$ .

$\frac{2 (3 + 0 \cdot 15)}{2 \cdot 2 + 10 \cdot 0}$

Step 3.10.2.3.5.2

Factor $2$ out of $10 \cdot 0$ .

$\frac{2 (3 + 0 \cdot 15)}{2 \cdot 2 + 2 (5 \cdot 0)}$

Step 3.10.2.3.5.3

Factor $2$ out of $2 (2) + 2 (5 \cdot 0)$ .

$\frac{2 (3 + 0 \cdot 15)}{2 (2 + 5 \cdot 0)}$

Step 3.10.2.3.5.4

Cancel the common factor.

$\frac{2 (3 + 0 \cdot 15)}{2 (2 + 5 \cdot 0)}$

Step 3.10.2.3.5.5

Rewrite the expression.

$\frac{3 + 0 \cdot 15}{2 + 5 \cdot 0}$

Step 3.10.2.4

Simplify the numerator.

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Step 3.10.2.4.1

Multiply $0$ by $15$ .

$\frac{3 + 0}{2 + 5 \cdot 0}$

Step 3.10.2.4.2

Add $3$ and $0$ .

$\frac{3}{2 + 5 \cdot 0}$

Step 3.10.2.5

Simplify the denominator.

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Step 3.10.2.5.1

Multiply $5$ by $0$ .

$\frac{3}{2 + 0}$

Step 3.10.2.5.2

Add $2$ and $0$ .

$\frac{3}{2}$

Step 4

List the horizontal asymptotes:

$y = \frac{3}{2}$

Step 5

There is no oblique asymptote because the degree of the numerator is less than or equal to the degree of the denominator.

No Oblique Asymptotes

Step 6

This is the set of all asymptotes.

No Vertical Asymptotes

Horizontal Asymptotes: $y = \frac{3}{2}$

No Oblique Asymptotes

Step 7