How to know if a rational function has either a vertical, horizontal or both asymptotes?

Question

1) Determine the horizontal and vertical asymptotes in case they exist from the graph of the function.

$f(x) = \frac{|x|+1}{x-2}$

By looking at the function I can't deduce what kind of asymptotes it has.

I checked the graph of this function with a program, and it has the 2 asymptotes.

Another thing I didn't understand, this $|x|$, does it mean the function has 2 horizontal asymptotes ? , because as I approach x from both + infinity and - infinity I can check in the graph that Y gets close to $-1$ and $1$.

Answer 1

Since we cannot factor out a $x-2$ term in the numerator for either of the $x \geq 0$ or $x < 0$ cases (the numerator is only a polynomial of degree $1$, after all), we can conclude that $x=2$ must be the sole vertical asymptote.

Now, as we discussed, the function $f(x)$ is defined to be piecewise rational as follows:

$$f(x) = \begin{cases} \frac{x+1}{x-2}, \quad x \geq 0 \\ \frac{-x+1}{x-2}, \quad x < 0 \\ \end{cases}$$

The definition of horizontal asymptote is as follows: we say that the function $f(x)$ has a horizontal asymptote $y = c$ if

$$\lim_{x \rightarrow +\infty} f(x) = c \quad \text{or} \quad \lim_{x \rightarrow -\infty} f(x) = c$$

The or part is what you should pay attention to, as we need only satisfy one of these conditions for $y = c$ to be a horizontal asymptote.

From this definition, we consider the piecewise definitions of $f(x)$ in each respective to domain to find the horizontal asymptotes. In your comment you said you can divide by the leading coefficient, but I think you meant to say that you divide by the leading variable (in this case, $x$) in both the numerator and denominator. This part is straightforward, and at once you can see that

$$y = 1 \quad \text{and} \quad y = -1$$

are indeed horizontal asymptotes as $x \rightarrow +\infty$ and $x \rightarrow -\infty$, respectively.

Answer 2

As noted in the comments your function is not rational, but only piecewise rational.

Anyway , a rational function of the form $$f(x)=\frac{p^n(x)}{q^m(x)}$$ where $m$ and $n$ are the degrees of the two polynomials, we have :

1) If $x=a$ is a root of $q^m(x)$ and it is not a root of $p^n(x)$ than $x=a $ is a vertical asymptote.

2) if $n<m$ than $y=0$ is an horizontal asymptote

3) if $n=m$ than the funtion has an horizontal asymptote $y=k\ne0$

4) if $n=m+1$ the function has an oblique asymptote.