# Thread: Domain, range, and equation for horizontal and vertical asymptotes?

1. ## Domain, range, and equation for horizontal and vertical asymptotes?

Can someone explain how to find those for:
y = 1 / (x^2 + 2) - 1

Correct me if I'm wrong, but the domain is all numbers X cannot be, the range is all numbers Y can be but I am unsure on how to find them and unsure of what asymptotes are and how to find those too.

2. Originally Posted by olen12
Can someone explain how to find those for:
y = 1 / (x^2 + 2) - 1

Correct me if I'm wrong, but the domain is all numbers X cannot be, the range is all numbers Y can be but I am unsure on how to find them and unsure of what asymptotes are and how to find those too.
Let's asume that
$
y=f(x)=\frac{1}{x^2+2}-1$

f is defined on R, because there isn't any x so
$
x^2+2=0$

$
\lim_{x->-\infty}f(x)=\lim_{x->\infty}f(x)=-1
$

$
f'(x)=-\frac{2x}{(x^2+2)^2}$

So x raises from -infinity to 0 and deacreases from 0 to +infinity
Which means that 0 is an local maxima
$
\lim_{x->0}f(x)=\frac{1}{2}
$

$
So f:R->(-1,\frac{1}{2}]
$

$
\lim_{x->-\infty}f(x)=\lim_{x->\infty}f(x)=-1
$

So f has y=-1 horizontal asymptote.

There isn't any real number a, so
$
\lim_{x->a}f(x)=\infty
or
\lim_{x->a}f(x)=-\infty
$

Which means f doesn't have vertical asymptote.
f doesen't have oblical asymptote neither.

3. Hi, thanks for the help but a few things I don't understand.

What does 'lim' mean?

And what is R?

4. Originally Posted by olen12
Hi, thanks for the help but a few things I don't understand.

What does 'lim' mean?

And what is R?
lim means the limit, or more specifically, the limit of a function as x approaches a particular value. R denotes the set of all real numbers. Saying that "the domain of the function is R" means "the function is defined for all real values of x."