Question

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

Answer 1

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

Answer 2

solve

x^2 + 1 = 0

for x

Answer 3

tell me what you get

Answer 4

Would it just be x^2 = -1 ?

Answer 5

good so far, what's next?

Answer 6

there will be no vertical asymptote

Answer 7

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

Answer 8

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

Answer 9

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

Answer 10

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

Answer 11

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

Answer 12

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

Answer 13

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

Answer 14

So basically the numerator was pointless in this equation?

Answer 15

pretty much

Answer 16

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

then look at the denominator only

Answer 17

Got it, thank you again.!

Answer 18

sure thing