Horizontal Asymptopes

[TheNextOne]

Can there only be 1 horizontal asymptope in a function?
Do lim (x approaching from - infinity) and lim (x approaching from + infinty) have to equal each other for there to be a horizontal asymptope?

 

[stapel]

Can there only be one horizontal asymptope in a function?

How does your text define "horizontal asymptote"? Are you considering only continuous functions, or are piecewise functions allowed?

Please reply with clarification, including your thoughts on the issue. Thank you.

Eliz.

 

[TheNextOne]

The text is considering only continous functions. I am just a little confused. I want to know wheter a continous function can have two seperate horizontal asymptopes. Also, does f'(0) = 0 imply that f(x) has a horizontal asymptope at y =0.

What does f'(X) = 0 if x <-1 or x> 1 mean?
Does this mean that f(x) has horizontal asymptopes at anything less than or graeter than -1 and 1?
Thank you.

 

[skeeter]

have a look at the function ...

\(\displaystyle y=\frac{x}{\sqrt{x^2+1}}\) ... what are its horizontal asymptotes?

If a function f(x) has a horizontal asymptote, then f'(x) approaches 0 as x->infinity and -infinity

 

[TheNextOne]

I think the horizontal aymptope for that function is y=1. I think I understand how to get horizontal asymptopes when given a function. I don't know what f'(x)= 0 when x<-1 or x>1 means. Could somebody please explain this to me?

I also have trouble drawing a graph when the funcion is given.

For example:

Sketch the graph of a positive continuous function y = f(x) having all of these properties:
absolute maximum point at (0, 4);
f’(x) > 0 if x= (-infinity, 0) U (2, + infinity)
f’(x) < 0 if x= (0,2)
f’’(x) < 0 if x= (-2,2) U (2,infinty)
f’’(x) > 0 if x= (-infinty,-2)
lim (as x approaches – infinty) f(x)=0
lim (as x approaches + infinty) F(x) = 3

This is what I get from the above question:

The absolute maximum occurs at (0,4).
Two horizontal asympopes ath y=0 and y=3.
The function is increasing from -infinty to 0 and from 2 to + infinity.
The function is decreasing from 0 to 2.
The function is concave down from -2 to 2 and from 2 to positive infinity.
The function is concave up from - infinty to -2.

I am having trouble drawing this and would appreciate it if somebody could post a solution for me to verify if i am drwing it correctly.

 

[stapel]

You have already posted the "graphing" exercise elsewhere. Please do not post the same exercise to multiple threads. Thank you.

Meanwhile, how does your text define "horizontal asymptote"? Please reply with a copy of whatever your book says. Thank you.

Eliz.