Trigonometry Examples

$f (x) = \frac{10 x (100 - x)}{25 + x}$

Step 1

Find where the expression $\frac{10 x (100 - x)}{25 + x}$ is undefined.

$x = - 25$

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 $m$ .

$n = 2$

$m = 1$

Step 4

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

No Horizontal Asymptotes

Step 5

Find the oblique asymptote using polynomial division.

Tap for more steps...

Step 5.1

Factor $10 x$ out of $1000 x - 10 x^{2}$ .

Tap for more steps...

Step 5.1.1

Factor $10 x$ out of $1000 x$ .

$\frac{10 x (100) - 10 x^{2}}{25 + x}$

Step 5.1.2

Factor $10 x$ out of $- 10 x^{2}$ .

$\frac{10 x (100) + 10 x (- x)}{25 + x}$

Step 5.1.3

Factor $10 x$ out of $10 x (100) + 10 x (- x)$ .

$\frac{10 x (100 - x)}{25 + x}$

Step 5.2

Expand $10 x (100 - x)$ .

Tap for more steps...

Step 5.2.1

Apply the distributive property.

$\frac{10 x \cdot 100 + 10 x (- x)}{25 + x}$

Step 5.2.2

Move $x$ .

$\frac{10 \cdot (100 x) + 10 x (- x)}{25 + x}$

Step 5.2.3

Remove parentheses.

$\frac{10 \cdot (100 x) + 10 x \cdot (- 1 x)}{25 + x}$

Step 5.2.4

Move $x$ .

$\frac{10 \cdot (100 x) + 10 \cdot (- 1 x \cdot x)}{25 + x}$

Step 5.2.5

Multiply $10$ by $100$ .

$\frac{1000 x + 10 \cdot (- 1 x \cdot x)}{25 + x}$

Step 5.2.6

Multiply $10$ by $- 1$ .

$\frac{1000 x - 10 x \cdot x}{25 + x}$

Step 5.2.7

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

$\frac{1000 x - 10 (x \cdot x)}{25 + x}$

Step 5.2.8

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

$\frac{1000 x - 10 (x \cdot x)}{25 + x}$

Step 5.2.9

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

$\frac{1000 x - 10 x^{1 + 1}}{25 + x}$

Step 5.2.10

Add $1$ and $1$ .

$\frac{1000 x - 10 x^{2}}{25 + x}$

Step 5.2.11

Reorder $1000 x$ and $- 10 x^{2}$ .

$\frac{- 10 x^{2} + 1000 x}{25 + x}$

Step 5.3

Reorder $25$ and $x$ .

$\frac{- 10 x^{2} + 1000 x}{x + 25}$

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 $0$ .

$x$

$25$

$10 x^{2}$

$1000 x$

$0$

Step 5.5

Divide the highest order term in the dividend $- 10 x^{2}$ by the highest order term in divisor $x$ .

				-	$10 x$
	$x$	+	$25$	-	$10 x^{2}$	+	$1000 x$	+	$0$

Step 5.6

Multiply the new quotient term by the divisor.

			-	$10 x$
$x$	+	$25$	-	$10 x^{2}$	+	$1000 x$	+	$0$
			-	$10 x^{2}$	-	$250 x$

Step 5.7

The expression needs to be subtracted from the dividend, so change all the signs in $- 10 x^{2} - 250 x$

			-	$10 x$
$x$	+	$25$	-	$10 x^{2}$	+	$1000 x$	+	$0$
			+	$10 x^{2}$	+	$250 x$

Step 5.8

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

			-	$10 x$
$x$	+	$25$	-	$10 x^{2}$	+	$1000 x$	+	$0$
			+	$10 x^{2}$	+	$250 x$
					+	$1250 x$

Step 5.9

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

			-	$10 x$
$x$	+	$25$	-	$10 x^{2}$	+	$1000 x$	+	$0$
			+	$10 x^{2}$	+	$250 x$
					+	$1250 x$	+	$0$

Step 5.10

Divide the highest order term in the dividend $1250 x$ by the highest order term in divisor $x$ .

			-	$10 x$	+	$1250$
$x$	+	$25$	-	$10 x^{2}$	+	$1000 x$	+	$0$
			+	$10 x^{2}$	+	$250 x$
					+	$1250 x$	+	$0$

Step 5.11

Multiply the new quotient term by the divisor.

			-	$10 x$	+	$1250$
$x$	+	$25$	-	$10 x^{2}$	+	$1000 x$	+	$0$
			+	$10 x^{2}$	+	$250 x$
					+	$1250 x$	+	$0$
					+	$1250 x$	+	$31250$

Step 5.12

The expression needs to be subtracted from the dividend, so change all the signs in $1250 x + 31250$

			-	$10 x$	+	$1250$
$x$	+	$25$	-	$10 x^{2}$	+	$1000 x$	+	$0$
			+	$10 x^{2}$	+	$250 x$
					+	$1250 x$	+	$0$
					-	$1250 x$	-	$31250$

Step 5.13

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

			-	$10 x$	+	$1250$
$x$	+	$25$	-	$10 x^{2}$	+	$1000 x$	+	$0$
			+	$10 x^{2}$	+	$250 x$
					+	$1250 x$	+	$0$
					-	$1250 x$	-	$31250$
							-	$31250$

Step 5.14

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

$- 10 x + 1250 - \frac{31250}{x + 25}$

Step 5.15

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

$y = - 10 x + 1250$

Step 6

This is the set of all asymptotes.

Vertical Asymptotes: $x = - 25$

No Horizontal Asymptotes

Oblique Asymptotes: $y = - 10 x + 1250$

Step 7