Calculus Examples

$y = x + \ln (| x |)$

Step 1

Find where the expression

$x + \ln (| x |)$ is undefined.

$x = 0$

Step 2

Ignoring the logarithm, 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

There are no horizontal asymptotes because

$Q (x)$ is

$1$ .

No Horizontal Asymptotes

Step 4

No oblique asymptotes are present for logarithmic and trigonometric functions.

No Oblique Asymptotes

Step 5

This is the set of all asymptotes.

Vertical Asymptotes:

$x = 0$

No Horizontal Asymptotes

Step 6