Can a Function Have a Horizontal Asymptote and Intersect It Infinitely?

[wuffle]

Homework Statement

2 problems.

1)
Find an example of a function f such that :
the line y=2 is a horizontal asymptote of the curve y=f(x)
the curve intersects the line y=2 at the infinitive number of points

2)
The position of an object moving along x-axis is given at time t by:
s(t)= 4t-4 if 2<t<8
and = -68 +t(20-t) if 8<=t<=10

Determine the acceleration and the velocity at any time t. Is the velocity continuous? Is acceleration continuous?

Homework Equations

The Attempt at a Solution

For the first one, I'm really puzzled how y=2 can be horizontal asymptote, AND that the curve intersects it at infinitive points.

I'm guessing that the function is y=2cos(1/x), as , y=2 is an asymptote even if my guess is right, how exactly should I solve that?

For the second one, I found derivatives of s and s'

s'= 4 if 2<t<8
s'= 20-2t if 8<=t<=10
s''=0 if 2<t<8 and
s"= -2 if 8<=t<=10

what then? how do i check if they are continuous?

I know the definition of continuous function and what exactly is horizontal asymptote, but I have no idea how to solve these two problems here :(.

Help!

 

[susskind_leon]

1. I guess your example is correct. You should use whatever definition of asymptote you have and see if it applies to your example. Just another, maybe slightly simpler example: 2+sin(x)/x

2. Just check if the functions "fit together" at t=8. For the velocity, you have 4 and 20-2*8=4, so the velocity is continuous. For the acceleration, you have 0 and -2, so the acceleration is non-continuous. In this case, continuity just means that you can draw the graph of the function without having to lift the pen at some point, so it's very intuitive. Check http://en.wikipedia.org/wiki/Piecewise

 

[wuffle]

1. Yeah I figured my example was correct, does my function pass the fact that

"the curve intersects the line y=2 at the infinitive number of points" though?

I just have no idea if it does or not.

2. I kinda figured that actually later onThank you for your answer!

 

[wuffle]

Sorry for bumping, I still am not really sure if I did #1 right, both
y=2cos(1/x) and y=2+sinx/x do seem to have y=2 as horizontal asymptotes, but I am not really sure how the curves intersects the line y=2 at the infinitive number of points, how can that be if the line y=2 is an asymptote?

 

[susskind_leon]

Ummm, I just noticed that the curve in your example does not intersect, but is rather tangent to y=2 infinitely many times. So, to be on the safe side, you should use my example.