Calculus Examples

$y = \frac{x^{3}}{64 - x^{2}}$

Step 1

Find where the expression $\frac{x^{3}}{64 - x^{2}}$ is undefined.

$x = - 8, x = 8$

Step 2

Since $\frac{x^{3}}{64 - x^{2}}$ $\to$ $\infty$ as $x$ $\to$ $- 8$ from the left and $\frac{x^{3}}{64 - x^{2}}$ $\to$ $- \infty$ as $x$ $\to$ $- 8$ from the right, then $x = - 8$ is a vertical asymptote.

$x = - 8$

Step 3

Since $\frac{x^{3}}{64 - x^{2}}$ $\to$ $\infty$ as $x$ $\to$ $8$ from the left and $\frac{x^{3}}{64 - x^{2}}$ $\to$ $- \infty$ as $x$ $\to$ $8$ from the right, then $x = 8$ is a vertical asymptote.

$x = 8$

Step 4

List all of the vertical asymptotes:

$x = - 8, 8$

Step 5

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 6

Find $n$ and $m$ .

$n = 3$

$m = 2$

Step 7

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

No Horizontal Asymptotes

Step 8

Find the oblique asymptote using polynomial division.

Tap for more steps...

Step 8.1

Simplify the denominator.

Tap for more steps...

Step 8.1.1

Rewrite $64$ as $8^{2}$ .

$\frac{x^{3}}{8^{2} - x^{2}}$

Step 8.1.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 8$ and $b = x$ .

$\frac{x^{3}}{(8 + x) (8 - x)}$

Step 8.2

Expand $(8 + x) (8 - x)$ .

Tap for more steps...

Step 8.2.1

Apply the distributive property.

$\frac{x^{3}}{8 (8 - x) + x (8 - x)}$

Step 8.2.2

Apply the distributive property.

$\frac{x^{3}}{8 \cdot 8 + 8 (- x) + x (8 - x)}$

Step 8.2.3

Apply the distributive property.

$\frac{x^{3}}{8 \cdot 8 + 8 (- x) + x \cdot 8 + x (- x)}$

Step 8.2.4

Remove parentheses.

$\frac{x^{3}}{8 \cdot 8 + 8 \cdot (- 1 x) + x \cdot 8 + x (- x)}$

Step 8.2.5

Reorder $x$ and $8$ .

$\frac{x^{3}}{8 \cdot 8 + 8 \cdot (- 1 x) + 8 \cdot x + x (- x)}$

Step 8.2.6

Remove parentheses.

$\frac{x^{3}}{8 \cdot 8 + 8 \cdot (- 1 x) + 8 \cdot x + x \cdot (- 1 x)}$

Step 8.2.7

Reorder $x$ and $- 1$ .

$\frac{x^{3}}{8 \cdot 8 + 8 \cdot (- 1 x) + 8 x - 1 \cdot (x \cdot x)}$

Step 8.2.8

Multiply $- 1$ by $x$ .

$\frac{x^{3}}{8 \cdot 8 + 8 \cdot (- 1 x) + 8 x - 1 x \cdot x}$

Step 8.2.9

Multiply $8$ by $8$ .

$\frac{x^{3}}{64 + 8 \cdot (- 1 x) + 8 x - 1 x \cdot x}$

Step 8.2.10

Multiply $8$ by $- 1$ .

$\frac{x^{3}}{64 - 8 x + 8 x - 1 x \cdot x}$

Step 8.2.11

Factor out negative.

$\frac{x^{3}}{64 - 8 x + 8 x - (x \cdot x)}$

Step 8.2.12

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

$\frac{x^{3}}{64 - 8 x + 8 x - (x \cdot x)}$

Step 8.2.13

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

$\frac{x^{3}}{64 - 8 x + 8 x - (x \cdot x)}$

Step 8.2.14

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

$\frac{x^{3}}{64 - 8 x + 8 x - x^{1 + 1}}$

Step 8.2.15

Add $1$ and $1$ .

$\frac{x^{3}}{64 - 8 x + 8 x - x^{2}}$

Step 8.2.16

Add $- 8 x$ and $8 x$ .

$\frac{x^{3}}{64 + 0 - x^{2}}$

Step 8.2.17

Subtract $x^{2}$ from $0$ .

$\frac{x^{3}}{64 - x^{2}}$

Step 8.2.18

Reorder $64$ and $- x^{2}$ .

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

Step 8.3

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

$x^{2}$

$0 x$

$64$

$x^{3}$

$0 x^{2}$

$0 x$

$0$

Step 8.4

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

$x$

$x^{2}$

$0 x$

$64$

$x^{3}$

$0 x^{2}$

$0 x$

$0$

Step 8.5

Multiply the new quotient term by the divisor.

$x$

$x^{2}$

$0 x$

$64$

$x^{3}$

$0 x^{2}$

$0 x$

$0$

$x^{3}$

$0$

$64 x$

Step 8.6

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

$x$

$x^{2}$

$0 x$

$64$

$x^{3}$

$0 x^{2}$

$0 x$

$0$

$x^{3}$

$0$

$64 x$

Step 8.7

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

$x$

$x^{2}$

$0 x$

$64$

$x^{3}$

$0 x^{2}$

$0 x$

$0$

$x^{3}$

$0$

$64 x$

Step 8.8

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

$x$

$x^{2}$

$0 x$

$64$

$x^{3}$

$0 x^{2}$

$0 x$

$0$

$x^{3}$

$0$

$64 x$

$0$

Step 8.9

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

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

Step 8.10

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

$y = - x$

Step 9

This is the set of all asymptotes.

Vertical Asymptotes: $x = - 8, 8$

No Horizontal Asymptotes

Oblique Asymptotes: $y = - x$

Step 10