please help... about sketching graphs

[furkang]

I want to ask you an exam question about sketching graph of a function, please help me;

I want you to look at question 2 from the following link especially 2 d)(about asymptotes)


question (http://www.math.metu.edu.tr/WWW/courses/math119/119F0405M2.pdf)

and the solutions given by the department are here;


solutions (http://www.math.metu.edu.tr/WWW/courses/math119/119F0405M2Sol.pdf)

what i did not understand is that; i think y=0 line must be another horizontal asymptote .Am ı wrong???

thank you for your helps...

[HallsofIvy]

I don't understand WHY you think that y=0 "must be another horizontal asymptotote".

In order for y= k to be a horizontal asymptote for any function, the graph must approach that line as x goes to either +infinity or -infinity. In this case, the derivative, to the right of x= 1 IS the constant -4 so the graph is the line y= -4.
There is no reason to think that the derivative approaches any constant, much less 0, as x goes to -infinity.

[furkang]

I looked at the description of horizontal asymptote , so you're right the derivative does not approach any constant but as x goes to - infinity the derivative is always less than zero and always increasing , I thought that could be counted as asymptote.

I have description of horizontal asymptote:

"the graph of f(x) has a horizontal asymptote y=L if
lim x-->infinity f(x)=L" is this true?

and in the question "lim x--> - infinity f(x)=0" is this wrong?

must the graph reach a constant as you said , to be a hor. asy.?

thank you for your help...