Question

How to find the asymptotes of `f(x)=2 - 3/x^2`?

Answer 1

Vertical asymptote is `x =0` or `y`-axis

Horizontal asymptote is `y=2`

No oblique asymptote.

Explanation:

An ASYMPTOTE is a line that approches a curve, but NEVER meets it.

To find the vertical asymptote , put the denominator = 0 (because 0 cannot divide any number) and solve. This is where the function cannot exist.

Given below is the step-by-step walk through:

The curve can never touch `x =0`, or the `y`-axis , thus making it the vertical asymptote.

To find the horizontal asymptote , compare the degree of the expressions in the numerator and the denominator.

First, lets re-write the expression so we have one a common denominator.

Now we can compare the degrees of the numerator and the denominaotr.

The degree of the numerator = 2 and the degree of the denominator = 2.
Since the degrees are equal, the horizontal asymptote

`y = ("numerator's leading coefficient"/"denominator's leading coefficient")`

`y = 2/1 => y = 2`

The oblique asymptote is a line of the form y = mx + c.
Oblique asymptote exists when the degree of numerator = degree of denominator + 1
Here, the degree of the numerator = degree of the denominator = 2.
Therefore, the given function has no oblique asymptotes.