Calculus Examples

$f (x) = \frac{x^{4} + 3 x^{3} - 2 x}{2 x^{2} + 1}$

Step 1

Find where the expression $\frac{x (x^{3} + 3 x^{2} - 2)}{2 x^{2} + 1}$ is undefined.

The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.

Step 2

The vertical asymptotes occur at areas of infinite discontinuity.

No Vertical Asymptotes

Step 3

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 4

Find $n$ and $m$ .

$n = 4$

$m = 2$

Step 5

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

No Horizontal Asymptotes

Step 6

Find the oblique asymptote using polynomial division.

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Step 6.1

Factor $x$ out of $x^{4} + 3 x^{3} - 2 x$ .

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Step 6.1.1

Factor $x$ out of $x^{4}$ .

$\frac{x \cdot x^{3} + 3 x^{3} - 2 x}{2 x^{2} + 1}$

Step 6.1.2

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

$\frac{x \cdot x^{3} + x (3 x^{2}) - 2 x}{2 x^{2} + 1}$

Step 6.1.3

Factor $x$ out of $- 2 x$ .

$\frac{x \cdot x^{3} + x (3 x^{2}) + x \cdot - 2}{2 x^{2} + 1}$

Step 6.1.4

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

$\frac{x (x^{3} + 3 x^{2}) + x \cdot - 2}{2 x^{2} + 1}$

Step 6.1.5

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

$\frac{x (x^{3} + 3 x^{2} - 2)}{2 x^{2} + 1}$

Step 6.2

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

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Step 6.2.1

Apply the distributive property.

$\frac{x (x^{3} + 3 x^{2}) + x \cdot - 2}{2 x^{2} + 1}$

Step 6.2.2

Apply the distributive property.

$\frac{x \cdot x^{3} + x (3 x^{2}) + x \cdot - 2}{2 x^{2} + 1}$

Step 6.2.3

Remove parentheses.

$\frac{x \cdot x^{3} + x \cdot (3 x^{2}) + x \cdot - 2}{2 x^{2} + 1}$

Step 6.2.4

Reorder $x$ and $3$ .

$\frac{x \cdot x^{3} + 3 \cdot (x \cdot x^{2}) + x \cdot - 2}{2 x^{2} + 1}$

Step 6.2.5

Reorder $x$ and $- 2$ .

$\frac{x \cdot x^{3} + 3 \cdot (x \cdot x^{2}) - 2 \cdot x}{2 x^{2} + 1}$

Step 6.2.6

Multiply $3$ by $x$ .

$\frac{x \cdot x^{3} + 3 x \cdot x^{2} - 2 x}{2 x^{2} + 1}$

Step 6.2.7

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

$\frac{x \cdot x^{3} + 3 x \cdot x^{2} - 2 x}{2 x^{2} + 1}$

Step 6.2.8

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

$\frac{x^{1 + 3} + 3 x \cdot x^{2} - 2 x}{2 x^{2} + 1}$

Step 6.2.9

Add $1$ and $3$ .

$\frac{x^{4} + 3 x \cdot x^{2} - 2 x}{2 x^{2} + 1}$

Step 6.2.10

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

$\frac{x^{4} + 3 (x \cdot x^{2}) - 2 x}{2 x^{2} + 1}$

Step 6.2.11

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

$\frac{x^{4} + 3 x^{1 + 2} - 2 x}{2 x^{2} + 1}$

Step 6.2.12

Add $1$ and $2$ .

$\frac{x^{4} + 3 x^{3} - 2 x}{2 x^{2} + 1}$

Step 6.3

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

$2 x^{2}$

$0 x$

$1$

$x^{4}$

$3 x^{3}$

$0 x^{2}$

$2 x$

$0$

Step 6.4

Divide the highest order term in the dividend $x^{4}$ by the highest order term in divisor $2 x^{2}$ .

$\frac{x^{2}}{2}$

$2 x^{2}$

$0 x$

$1$

$x^{4}$

$3 x^{3}$

$0 x^{2}$

$2 x$

$0$

Step 6.5

Multiply the new quotient term by the divisor.

$\frac{x^{2}}{2}$

$2 x^{2}$

$0 x$

$1$

$x^{4}$

$3 x^{3}$

$0 x^{2}$

$2 x$

$0$

$x^{4}$

$0$

$\frac{x^{2}}{2}$

Step 6.6

The expression needs to be subtracted from the dividend, so change all the signs in $x^{4} + 0 + \frac{x^{2}}{2}$

$\frac{x^{2}}{2}$

$2 x^{2}$

$0 x$

$1$

$x^{4}$

$3 x^{3}$

$0 x^{2}$

$2 x$

$0$

$x^{4}$

$0$

$\frac{x^{2}}{2}$

Step 6.7

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

$\frac{x^{2}}{2}$

$2 x^{2}$

$0 x$

$1$

$x^{4}$

$3 x^{3}$

$0 x^{2}$

$2 x$

$0$

$x^{4}$

$0$

$\frac{x^{2}}{2}$

$3 x^{3}$

$\frac{x^{2}}{2}$

Step 6.8

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

$\frac{x^{2}}{2}$

$2 x^{2}$

$0 x$

$1$

$x^{4}$

$3 x^{3}$

$0 x^{2}$

$2 x$

$0$

$x^{4}$

$0$

$\frac{x^{2}}{2}$

$3 x^{3}$

$\frac{x^{2}}{2}$

$2 x$

Step 6.9

Divide the highest order term in the dividend $3 x^{3}$ by the highest order term in divisor $2 x^{2}$ .

$\frac{x^{2}}{2}$

$\frac{3 x}{2}$

$2 x^{2}$

$0 x$

$1$

$x^{4}$

$3 x^{3}$

$0 x^{2}$

$2 x$

$0$

$x^{4}$

$0$

$\frac{x^{2}}{2}$

$3 x^{3}$

$\frac{x^{2}}{2}$

$2 x$

Step 6.10

Multiply the new quotient term by the divisor.

$\frac{x^{2}}{2}$

$\frac{3 x}{2}$

$2 x^{2}$

$0 x$

$1$

$x^{4}$

$3 x^{3}$

$0 x^{2}$

$2 x$

$0$

$x^{4}$

$0$

$\frac{x^{2}}{2}$

$3 x^{3}$

$\frac{x^{2}}{2}$

$2 x$

$3 x^{3}$

$0$

$\frac{3 x}{2}$

Step 6.11

The expression needs to be subtracted from the dividend, so change all the signs in $3 x^{3} + 0 + \frac{3 x}{2}$

$\frac{x^{2}}{2}$

$\frac{3 x}{2}$

$2 x^{2}$

$0 x$

$1$

$x^{4}$

$3 x^{3}$

$0 x^{2}$

$2 x$

$0$

$x^{4}$

$0$

$\frac{x^{2}}{2}$

$3 x^{3}$

$\frac{x^{2}}{2}$

$2 x$

$3 x^{3}$

$0$

$\frac{3 x}{2}$

Step 6.12

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

$\frac{x^{2}}{2}$

$\frac{3 x}{2}$

$2 x^{2}$

$0 x$

$1$

$x^{4}$

$3 x^{3}$

$0 x^{2}$

$2 x$

$0$

$x^{4}$

$0$

$\frac{x^{2}}{2}$

$3 x^{3}$

$\frac{x^{2}}{2}$

$2 x$

$3 x^{3}$

$0$

$\frac{3 x}{2}$

$\frac{x^{2}}{2}$

$\frac{7 x}{2}$

Step 6.13

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

$\frac{x^{2}}{2}$

$\frac{3 x}{2}$

$2 x^{2}$

$0 x$

$1$

$x^{4}$

$3 x^{3}$

$0 x^{2}$

$2 x$

$0$

$x^{4}$

$0$

$\frac{x^{2}}{2}$

$3 x^{3}$

$\frac{x^{2}}{2}$

$2 x$

$3 x^{3}$

$0$

$\frac{3 x}{2}$

$\frac{x^{2}}{2}$

$\frac{7 x}{2}$

$0$

Step 6.14

Divide the highest order term in the dividend $- \frac{x^{2}}{2}$ by the highest order term in divisor $2 x^{2}$ .

$\frac{x^{2}}{2}$

$\frac{3 x}{2}$

$\frac{1}{4}$

$2 x^{2}$

$0 x$

$1$

$x^{4}$

$3 x^{3}$

$0 x^{2}$

$2 x$

$0$

$x^{4}$

$0$

$\frac{x^{2}}{2}$

$3 x^{3}$

$\frac{x^{2}}{2}$

$2 x$

$3 x^{3}$

$0$

$\frac{3 x}{2}$

$\frac{x^{2}}{2}$

$\frac{7 x}{2}$

$0$

Step 6.15

Multiply the new quotient term by the divisor.

						$\frac{x^{2}}{2}$	+	$\frac{3 x}{2}$	-	$\frac{1}{4}$
$2 x^{2}$	+	$0 x$	+	$1$		$x^{4}$	+	$3 x^{3}$	+	$0 x^{2}$	-	$2 x$	+	$0$
					-	$x^{4}$	-	$0$	-	$\frac{x^{2}}{2}$
							+	$3 x^{3}$	-	$\frac{x^{2}}{2}$	-	$2 x$
							-	$3 x^{3}$	-	$0$	-	$\frac{3 x}{2}$
									-	$\frac{x^{2}}{2}$	-	$\frac{7 x}{2}$	+	$0$
									-	$\frac{x^{2}}{2}$	+	$0$	-	$\frac{1}{4}$

Step 6.16

The expression needs to be subtracted from the dividend, so change all the signs in $- \frac{x^{2}}{2} + 0 - \frac{1}{4}$

$\frac{x^{2}}{2}$

$\frac{3 x}{2}$

$\frac{1}{4}$

$2 x^{2}$

$0 x$

$1$

$x^{4}$

$3 x^{3}$

$0 x^{2}$

$2 x$

$0$

$x^{4}$

$0$

$\frac{x^{2}}{2}$

$3 x^{3}$

$\frac{x^{2}}{2}$

$2 x$

$3 x^{3}$

$0$

$\frac{3 x}{2}$

$\frac{x^{2}}{2}$

$\frac{7 x}{2}$

$0$

$\frac{x^{2}}{2}$

$0$

$\frac{1}{4}$

Step 6.17

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

$\frac{x^{2}}{2}$

$\frac{3 x}{2}$

$\frac{1}{4}$

$2 x^{2}$

$0 x$

$1$

$x^{4}$

$3 x^{3}$

$0 x^{2}$

$2 x$

$0$

$x^{4}$

$0$

$\frac{x^{2}}{2}$

$3 x^{3}$

$\frac{x^{2}}{2}$

$2 x$

$3 x^{3}$

$0$

$\frac{3 x}{2}$

$\frac{x^{2}}{2}$

$\frac{7 x}{2}$

$0$

$\frac{x^{2}}{2}$

$0$

$\frac{1}{4}$

$\frac{7 x}{2}$

$\frac{1}{4}$

Step 6.18

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

$\frac{x^{2}}{2} + \frac{3 x}{2} - \frac{1}{4} + \frac{- \frac{7 x}{2} + \frac{1}{4}}{2 x^{2} + 1}$

Step 6.19

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

$y = \frac{x^{2}}{2} + \frac{3 x}{2} - \frac{1}{4}$

Step 7

This is the set of all asymptotes.

No Vertical Asymptotes

No Horizontal Asymptotes

Oblique Asymptotes: $y = \frac{x^{2}}{2} + \frac{3 x}{2} - \frac{1}{4}$

Step 8