Precalculus Examples

$y = \frac{(x - 7) (x - 3)}{x^{2} - 10 x + 21}$

Step 1

Find where the expression $\frac{(x - 7) (x - 3)}{x^{2} - 10 x + 21}$ is undefined.

$x = 3, x = 7$

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 $m$ .

$n = 2$

$m = 2$

Step 5

Since $n = m$ , the horizontal asymptote is the line $y = \frac{a}{b}$ where $a = 1$ and $b = 1$ .

$y = 1$

Step 6

$y = \frac{(x - 7) (x - 3)}{x^{2} - 10 x + 21}$ is an equation of a line, which means there are no oblique asymptotes.

No Oblique Asymptotes

Step 7

This is the set of all asymptotes.

No Vertical Asymptotes

Horizontal Asymptotes: $y = 1$

No Oblique Asymptotes

Step 8