Question

How do you tell if it's a vertical asymptote function or a horizontal asymptote function?

Answer 1

See explanation

Explanation:

To see if a function has vertical asymptote you have to find values of `x` which are not in the domain, but their surrounding is. For example if `f(x)=1/x`, then `x=0` is a vertical asymptote. To ensure that such point is an asymptote you have to calculate left and right side limits:

`lim_{x->0^+}1/x=+oo`

`lim_{x->0^-}1/x=-oo`

graph{(1/x) [-10, 10, -5, 5]}

To find if a function has horizontal (or oblique) asymptotes you have to calculate limits:

`a= lim_{x->-oo} f(x)/x` and `b=lim_{x->-oo} f(x)-ax`

If both limits are finite, then function `f(x)=ax+b` is an oblique (or horizontal for `a=0`) asymptote for `f(x)`.
For example function `f(x)=x+1/x` has an oblique asymptote `f(x)=x`

graph{(y-x-1/x)(y-x)=0 [-10, 10, -5, 5]}

Answer 2

See the explanation.

Explanation:

Vertical

`x=h` is a vertical asymptote if `f(x)` increases or decreases without bound (`f(x)rarroo" or "-oo`) as `x` approaches `h` from at least one side.

Roughly: A vertical asymptote occurs if `y` goes to `+-` infinity as `x` is approaching some finite number

Horizontal

Roughly: A horizontal asymptote occurs if `y` goes to some finite number as `x` is goes to `+-` infinity.

`y=k` is a horizontal asymptote if `f(x) rarr k` as `x` increases or decreases without bound (`x rarr oo" or "-oo`)