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

• April 15th 2009, 01:05 PM
olen12
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.
• April 15th 2009, 03:12 PM
m3th0dman
Quote:

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.
• April 15th 2009, 03:45 PM
olen12
Hi, thanks for the help but a few things I don't understand.

What does 'lim' mean?

And what is R?
• April 15th 2009, 03:54 PM
icemanfan
Quote:

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."