How to determine asymptotes?

Question

In general do we say that:

1. If, for some given $x$, we get $y = 0/0$, it's a removable discontinuity.

2. If, for some given $x$, we get $y = a/0$ for $a \neq 0$, it's a vertical asymptote at that $x$.

3. If, for $x \to \infty$, we get $y = c$, then there is a horizontal asymptote at $y=c$.

Would you say this is correct so far? Am I missing anything?

Answer 1

Point 1 is not true in general, for point 2 the case is $f(x)=\frac{g(x)}{h(x)}$ with $h(x_0)=0$ and $g(x_0)\neq 0$ and x_0 cluster point, point 3 are correct.

Remember also the case for oblique asymptotes

• $y=mx+n$ with $m,n\in \mathbb{R}$.

with

$$m=\lim_{x\rightarrow+\infty}\frac{f(x)}{x}$$

and

$$n=\lim_{x\rightarrow+\infty} (f(x)-mx)$$

when both limit exist.

Answer 2

For $(1)$ ,$0/0$ is not always a removable discontinuity.

For $(3)$ , you may add $x\to -\infty $ as well.

I do not see any other issues.