Calculus Examples

$64 x^{2} + \frac{54}{x} + 2$

Step 1

Find where the expression $64 x^{2} + \frac{54}{x} + 2$ is undefined.

$x = 0$

Step 2

Consider the rational function $R (x) = \frac{a x^{n}}{b x^{m}}$ where $n$ is the degree of the numerator and $m$ is the degree of the denominator.

1. If $n < m$ , then the x-axis, $y = 0$ , is the horizontal asymptote.

2. If $n = m$ , then the horizontal asymptote is the line $y = \frac{a}{b}$ .

3. If $n > m$ , then there is no horizontal asymptote (there is an oblique asymptote).

Step 3

Find $n$ and $m$ .

$n = 3$

$m = 1$

Step 4

Since $n > m$ , there is no horizontal asymptote.

No Horizontal Asymptotes

Step 5

Find the oblique asymptote using polynomial division.

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Step 5.1

Combine.

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Step 5.1.1

To write $64 x^{2}$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

$\frac{64 x^{2} x}{x} + \frac{54}{x} + 2$

Step 5.1.2

Combine the numerators over the common denominator.

$\frac{64 x^{2} x + 54}{x} + 2$

Step 5.1.3

Simplify the numerator.

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Step 5.1.3.1

Factor $2$ out of $64 x^{2} x + 54$ .

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Step 5.1.3.1.1

Factor $2$ out of $64 x^{2} x$ .

$\frac{2 (32 x^{2} x) + 54}{x} + 2$

Step 5.1.3.1.2

Factor $2$ out of $54$ .

$\frac{2 (32 x^{2} x) + 2 (27)}{x} + 2$

Step 5.1.3.1.3

Factor $2$ out of $2 (32 x^{2} x) + 2 (27)$ .

$\frac{2 (32 x^{2} x + 27)}{x} + 2$

Step 5.1.3.2

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

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Step 5.1.3.2.1

Move $x$ .

$\frac{2 (32 (x \cdot x^{2}) + 27)}{x} + 2$

Step 5.1.3.2.2

Multiply $x$ by $x^{2}$ .

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Step 5.1.3.2.2.1

Raise $x$ to the power of $1$ .

$\frac{2 (32 (x^{1} x^{2}) + 27)}{x} + 2$

Step 5.1.3.2.2.2

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

$\frac{2 (32 x^{1 + 2} + 27)}{x} + 2$

Step 5.1.3.2.3

Add $1$ and $2$ .

$\frac{2 (32 x^{3} + 27)}{x} + 2$

Step 5.1.4

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

$\frac{2 (32 x^{3} + 27)}{x} + \frac{2 x}{x}$

Step 5.1.5

Combine the numerators over the common denominator.

$\frac{2 (32 x^{3} + 27) + 2 x}{x}$

Step 5.1.6

Simplify the numerator.

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Step 5.1.6.1

Factor $2$ out of $2 (32 x^{3} + 27) + 2 x$ .

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Step 5.1.6.1.1

Factor $2$ out of $2 x$ .

$\frac{2 (32 x^{3} + 27) + 2 (x)}{x}$

Step 5.1.6.1.2

Factor $2$ out of $2 (32 x^{3} + 27) + 2 (x)$ .

$\frac{2 (32 x^{3} + 27 + x)}{x}$

Step 5.1.6.2

Reorder terms.

$\frac{2 (32 x^{3} + x + 27)}{x}$

Step 5.1.7

Simplify.

$\frac{64 x^{3} + 2 x + 54}{x}$

Step 5.2

Factor $2$ out of $64 x^{3} + 2 x + 54$ .

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Step 5.2.1

Factor $2$ out of $64 x^{3}$ .

$\frac{2 (32 x^{3}) + 2 x + 54}{x}$

Step 5.2.2

Factor $2$ out of $2 x$ .

$\frac{2 (32 x^{3}) + 2 (x) + 54}{x}$

Step 5.2.3

Factor $2$ out of $54$ .

$\frac{2 (32 x^{3}) + 2 x + 2 \cdot 27}{x}$

Step 5.2.4

Factor $2$ out of $2 (32 x^{3}) + 2 x$ .

$\frac{2 (32 x^{3} + x) + 2 \cdot 27}{x}$

Step 5.2.5

Factor $2$ out of $2 (32 x^{3} + x) + 2 \cdot 27$ .

$\frac{2 (32 x^{3} + x + 27)}{x}$

Step 5.3

Expand $2 (32 x^{3} + x + 27)$ .

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Step 5.3.1

Apply the distributive property.

$\frac{2 (32 x^{3} + x) + 2 \cdot 27}{x}$

Step 5.3.2

Apply the distributive property.

$\frac{2 (32 x^{3}) + 2 x + 2 \cdot 27}{x}$

Step 5.3.3

Remove parentheses.

$\frac{2 \cdot (32 x^{3}) + 2 x + 2 \cdot 27}{x}$

Step 5.3.4

Multiply $2$ by $32$ .

$\frac{64 x^{3} + 2 x + 2 \cdot 27}{x}$

Step 5.3.5

Multiply $2$ by $27$ .

$\frac{64 x^{3} + 2 x + 54}{x}$

Step 5.4

Set up the polynomials to be divided. If there is not a term for every exponent, insert one with a value of $0$ .

$x$

$0$

$64 x^{3}$

$0 x^{2}$

$2 x$

$54$

Step 5.5

Divide the highest order term in the dividend $64 x^{3}$ by the highest order term in divisor $x$ .

					$64 x^{2}$
	$x$	+	$0$		$64 x^{3}$	+	$0 x^{2}$	+	$2 x$	+	$54$

Step 5.6

Multiply the new quotient term by the divisor.

				$64 x^{2}$
$x$	+	$0$		$64 x^{3}$	+	$0 x^{2}$	+	$2 x$	+	$54$
			+	$64 x^{3}$	+	$0$

Step 5.7

The expression needs to be subtracted from the dividend, so change all the signs in $64 x^{3} + 0$

				$64 x^{2}$
$x$	+	$0$		$64 x^{3}$	+	$0 x^{2}$	+	$2 x$	+	$54$
			-	$64 x^{3}$	-	$0$

Step 5.8

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

				$64 x^{2}$
$x$	+	$0$		$64 x^{3}$	+	$0 x^{2}$	+	$2 x$	+	$54$
			-	$64 x^{3}$	-	$0$
						$0$

Step 5.9

Pull the next term from the original dividend down into the current dividend.

				$64 x^{2}$
$x$	+	$0$		$64 x^{3}$	+	$0 x^{2}$	+	$2 x$	+	$54$
			-	$64 x^{3}$	-	$0$
						$0$	+	$2 x$	+	$54$

Step 5.10

Divide the highest order term in the dividend $2 x$ by the highest order term in divisor $x$ .

				$64 x^{2}$	+	$0 x$	+	$2$
$x$	+	$0$		$64 x^{3}$	+	$0 x^{2}$	+	$2 x$	+	$54$
			-	$64 x^{3}$	-	$0$
						$0$	+	$2 x$	+	$54$

Step 5.11

Multiply the new quotient term by the divisor.

				$64 x^{2}$	+	$0 x$	+	$2$
$x$	+	$0$		$64 x^{3}$	+	$0 x^{2}$	+	$2 x$	+	$54$
			-	$64 x^{3}$	-	$0$
						$0$	+	$2 x$	+	$54$
							+	$2 x$	+	$0$

Step 5.12

The expression needs to be subtracted from the dividend, so change all the signs in $2 x + 0$

				$64 x^{2}$	+	$0 x$	+	$2$
$x$	+	$0$		$64 x^{3}$	+	$0 x^{2}$	+	$2 x$	+	$54$
			-	$64 x^{3}$	-	$0$
						$0$	+	$2 x$	+	$54$
							-	$2 x$	-	$0$

Step 5.13

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

				$64 x^{2}$	+	$0 x$	+	$2$
$x$	+	$0$		$64 x^{3}$	+	$0 x^{2}$	+	$2 x$	+	$54$
			-	$64 x^{3}$	-	$0$
						$0$	+	$2 x$	+	$54$
							-	$2 x$	-	$0$
									+	$54$

Step 5.14

The final answer is the quotient plus the remainder over the divisor.

$64 x^{2} + 2 + \frac{54}{x}$

Step 5.15

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

$y = 64 x^{2} + 2$

Step 6

This is the set of all asymptotes.

Vertical Asymptotes: $x = 0$

No Horizontal Asymptotes

Oblique Asymptotes: $y = 64 x^{2} + 2$

Step 7