## gjhfdfg 2 years ago Find the vertical asymptotes, if any, of the graph of the rational function.

1. gjhfdfg

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

2. jim_thompson5910

solve x^2 + 1 = 0 for x

3. jim_thompson5910

tell me what you get

4. gjhfdfg

Would it just be x^2 = -1 ?

5. jim_thompson5910

good so far, what's next?

6. k.rajabhishek

there will be no vertical asymptote

7. gjhfdfg

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

8. 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

9. jim_thompson5910

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

10. jim_thompson5910

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

11. gjhfdfg

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

12. 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

13. 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

14. gjhfdfg

So basically the numerator was pointless in this equation?

15. jim_thompson5910

pretty much

16. 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

17. gjhfdfg

Got it, thank you again.!

18. jim_thompson5910

sure thing