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

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neela | High School Teacher | (Level 3) Valedictorian

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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.


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