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- anonymous

Find the vertical asymptotes, if any, of the graph of the rational function.

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- anonymous

Find the vertical asymptotes, if any, of the graph of the rational function.

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- anonymous

f(x)= \[\frac{ x }{ x^2 +1 }\]

- jim_thompson5910

solve
x^2 + 1 = 0
for x

- jim_thompson5910

tell me what you get

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- anonymous

Would it just be x^2 = -1 ?

- jim_thompson5910

good so far, what's next?

- anonymous

there will be no vertical asymptote

- anonymous

Hmm, Im not sure? Do I replace the variable with -1?

- jim_thompson5910

take the square root of both sides
x = sqrt(-1) or x = -sqrt(-1)
but there's a problem, you can't take the square root of -1 and get a real number

- jim_thompson5910

so x^2 + 1 = 0 has no real solutions

- jim_thompson5910

leading to the fact that \[\Large \frac{ x }{ x^2 +1 }\] has no vertical asymptotes

- anonymous

Ah, ok. Thank you.!
Just out of curiosity, what happened to the x on top of the fraction?

- jim_thompson5910

the numerator doesn't play any role in finding the vertical asymptotes unless you can make it cancel with something in the denominator

- jim_thompson5910

if you had something like
x
--------
x^2+x
the fraction would simplify to
1
________
x + 1
and this would be a case where the numerator plays a role

- anonymous

So basically the numerator was pointless in this equation?

- jim_thompson5910

pretty much

- jim_thompson5910

the basic thing is to simplify as much as possible (which couldn't be done in this case)
then look at the denominator only

- anonymous

Got it, thank you again.!

- jim_thompson5910

sure thing

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