Wolfram Alpha:

```Asymptotes
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Vertical asymptotes are vertical lines which correspond to the zeroes of the
denominator of a rational function.

Horizontal asymptotes indicate general behavior far off to the sides of the
graph. The rules for finding horizontal asymptotes are:

f(x) = a(x)/b(x)

f(x) has vertical asymptote when b(x) = 0.

-  If the degree of a(x) is greater than the degree of b(x) there is no
horizontal asymptote. If the degree of a(x) is greater than b(x) by 1
there is most likely a slant (aka oblique) asymptote.  The equation for the
slant asymptote is the polynomial part of the result after performing the long
division (see Example 4.).

-  If the degree of the numerator is equal to the degree of the denominator,

-  If the degree of the numerator is less than the degree of the denominator,
the horizontal asymptote is the line y = 0.

Ex 1.

f(x) = (x2 + x - 2)/(x2 - x - 6)

f(x) = (x - 1)(x + 2)/(x + 2)(x - 3)   x ≠ -2

f(x) = (x - 1)/(x - 3)

vertical asymptote: x = 3

horizontal asymptote: y = 1

Ex 2.

f(x) = (-x2 - x + 2)/(x2 - x - 6)

f(x) = -(x - 1)(x + 2)/(x + 2)(x - 3)   x ≠ -2

f(x) = -(x - 1)/(x - 3)

vertical asymptote: x = 3

horizontal asymptote: y = -1

Ex 3.

f(x) = (x-3)/(x2 - x - 6)

f(x) = (x-3)/(x+2)(x-3)  x ≠ 3

vertical aymptote: x = -2

horizontal asymptote: y = 0

Ex 4.

y = x2 + 3x + 2/x - 2

=> x + 5 + 12/(x - 2)

The slant asymptote is the polynomial part of the answer.  So,

y = x + 5

and the vertical asymptote is x = 2

```