Vertical asymptotes are
vertical lines which correspond to the zeroes of the denominator of a
rational function.
(They can also arise in other contexts, such as logarithms, but you'll
almost certainly first encounter asymptotes in the context of rationals.) I'll give you an example:
This is a rational function.
More to the point, this is a fraction. Can you have a zero in the denominator
of a fraction? No. So if I set the denominator of the above fraction equal
to zero and solve, this will tell me the values that x
cannot be:
x2
– 5x – 6 = 0
(x
– 6)(x + 1) = 0
x
= 6 or –1
So x
cannot be
6 or –1,
because then I'd be dividing by zero.
The domain is the set
of all x-values
that I'm allowed to use. The only values that could be disallowed are
those that give me a zero in the denominator. So I'll set the denominator
equal to zero and solve.
x2
+ 2x – 8 = 0
(x
+ 4)(x – 2) = 0
x
= –4 or x
= 2
Since I can't have a
zero in the denominator, then I can't have x
= –4 or x
= 2 in the domain.
This tells me that the vertical asymptotes (which tell me where the
graph can not
go) will be at the
values x
= –4 or x
= 2.
domain:
vertical
asymptotes: x
= –4,
2
