Question

How would I find the Vertical and Horizontal Asymptotes of the following, using Limits?

Answer 1

\(\ \Huge y=\frac{x}{x-2}. \)

Answer 2

Also, How would I know whether to approach the HA from above or below for \(\ x=\pm\infty \)?

Answer 3

When Graphing

Answer 4

horizontal asymptotes are defined as the limit of a function, L, as x approaches inf and negative inf.

Answer 5

you'll have to look at both postive infinity and negative infinity... you might have two horizontal asymptotes..

Answer 6

Also, I'm a bit confused on finding the x/y intercepts.... Where do you plug in zero?

Answer 7

for vertical asymptotes, it's just the "opposite".... that is, take the limit of x/(x-2) as x approaches 2. if the limit approaches infinity or negative infinity, then x=2 is your vertical asymptote...

Answer 8

to find:
x intercepts, plug in a 0 for y and solve for x;
y intercepts, plug in a 0 for x and solve for y.

Answer 9

@dpalnc, what about my graphing question?

Answer 10

these question are all the things you need to create the graph...

Answer 11

I know but how do I know whether to approach the HA from above or below

Answer 12

first, do you know the limit of x/(x-2) as x approaches infinity...

Answer 13

Well, as it approaches + infinity, The Limit is one, and as it approaches negative infinity it also is one

Answer 14

ok so we know it only has 1 horizontal asymptote.
now tell me... is there a solution to this equation:

1 = x/(x-2)
??

Answer 15

look at the big picture... in order for that to be true, the numerator has to equal the denominator.. will that ever occur?

Answer 16

No?

Answer 17

correct... do you know what that means?

Answer 18

Not really

Answer 19

it means our graph will not cross the horizontal asymptote (yes, horizontal asymptotes can be crossed).

which means the graph will either be coming from the top and approach the asymptote or coming from the bottom and approach the asymptote...

Answer 20

still with me?

Answer 21

I think so

Answer 22

now, this is where our x/y intercepts come in.... what are they?

Answer 23

x & y intercepts are both (0,0)

Answer 24

ok..
what about our vertical asymptote(s)?

Answer 25

x=2

Answer 26

this is what we have so far to construct the graph without the actual graph itself.
but we actually have enough info to make a sketch of the function..
start with the x,y intercept...