Extended (3unit) Curve Sketching (1 Viewer)

[Slidey]

oblique asymptote is a slanted line

to find out if its oblique or horiz:
u find the limiting value of y as x approaches infinity
if the limiting value is a constant (number) then the asymptote is horizontal
if the limiting value is infinity or negative infinity then it's oblique (or not oblique, but not horizontal either)

to find the oblique asymptote:

->
you can divide, for example:
y = (x³+x²) / x²
y = x³/x² + x²/x²
y = x + 1
so asymptote is y=x+1

->
take limit, for example:
y = x + 1/x
as x-> infinity, the 1/x part approaches zero
so as x-> infinity, y=x
then asymptote is y=x

quote=blackfriday : "oblique asymptotes usually occur when the highest power in the numerator is higher than the highest power in the denominator."
higher by 1

Um... for y=(x^3+x^2)/x^2, the actualy curve IS y=x+1, that's not the asymptote. No asymptotes exist, however, a discontinuity at x=0 does.

So tell me the asymptote for y=(x^3+1)/x, besides a vertical at x=0. Does it have any others? Is perhaps y=x^2 an asymptote? I have a good feeling it is.

And what do you call an oblique asymptote which the curve actually cuts?

 

[mojako]

Um... for y=(x^3+x^2)/x^2, the actualy curve IS y=x+1, that's not the asymptote. No asymptotes exist, however, a discontinuity at x=0 does.

So tell me the asymptote for y=(x^3+1)/x, besides a vertical at x=0. Does it have any others? Is perhaps y=x^2 an asymptote? I have a good feeling it is.

And what do you call an oblique asymptote which the curve actually cuts?

LOL
another bad example from mojako

y=(x^3+1)/x
y=x^2 + 1/x
so y=x^2 is what the curve would behave far on the right and left of the coordinate axis, but I'm not really sure if it's called asymptote. when I went to dictionary.com it mentions a "line".

Also it defines asymptote as "A line whose distance to a given curve tends to zero. An asymptote may or may not intersect its associated curve."

now I need to think of an example to replace the y=(x^3+x^2)/x^2 one
hmm.... this will do: y = (x^3+x^2+7) / x^2

And a harder example for those who want it:
y = (x^2+x) / (2-x)
y = (x^2+x-6+6) / (2-x)
y = (2-x)(-x-3) / (2-x) + 6 / (2-x)

 

[blackfriday]

i think the best term to use for those sorts of restrictions is that it is 'asymptotic'

 

[JamiL]

when x can not equal sumthin it is vertical, and other asym must be foun by finding the imit of the function as x approches +or- infinity