Identify any vertical,horizontal or slant asymptotes for the function f(x)=x/x^2-1 And Graph the function

f(x) = x/x^2-1

To identify
the asymptotes.

Denominator x^2-1 =
(x-1)(x+1)

For x > 0,  f(x) = x/(x^2-1) > 0
and Lt x/x^2-1 = 0 as x--> 0.

Similarly , when 
x< -1,  x/(x^2-1) < 0 as numerator is negative and denominator is
positive. Therefore Lt x/(x^2-1)  as x--> -
infinity

Therefore x axis is an
asymptote.

When x = 1,  f(x) does not exist. But  Right Lt
x/(x^2-1) = infinity as x --> 1+

Left Lt x/(x^2-1) =
-infinity as x --> 1-

Therefore x =1 is a
vertical asyptote.

At x =-1  the f(x) = x /(x^2-1 doenot
exist as Rt lt x/(x^2-1) is infinity and left lt x/(x^2-1 is - infinity. Thus  x= -1 is
another vertical  asymptote.