Question

How do I find if the graph lies above or below an oblique asymptote?

Answer 1

plot a point!!

Answer 2

how though?

Answer 3

?
pick an x find the y plot the point see if it above or below the line

Answer 4

you got a specific example?

Answer 5

but how would i know if its above or below the line?

Answer 6

look!

Answer 7

yeah, (x^2+1)/x

Answer 8

but what if i am required to sketch it?

Answer 9

suppose you find \((2,3)\) on the graph of your function and the line is \(y=2x\) then on the line you would have \((2,4)\) which is evidently above \((2,3)\) so the curve would be below the line

Answer 10

ohh... okay. So we are comparing the vertical distances of the two points, right?

Answer 11

hold the phone
your function is \(\frac{x^2+1}{x}=x+\frac{1}{x}\) right?

Answer 12

yes

Answer 13

so the "oblique asymptote is \(y=x\)

Answer 14

yes

Answer 15

then it should be pretty clear without plotting a point that for \(x>0\) that \(x+\frac{1}{x}>x\) right?

Answer 16

and similarly for \(x<0\) you would have \(x+\frac{1}{x}<x\)

Answer 17

ok

Answer 18

so to the right of the vertical asymptote at \(x=0\) the curve lies above the asymptote, and to the left of \(x=0\) it lies below the asymptote

Answer 19

i mean of course above the oblique asymptote

Answer 20

got it!! thank you so much :)