# How to know if this irrational function has no asymptotes?

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1. Sep 18, 2015

### Jeanclaud

1. The problem statement, all variables and given/known dat
F(x)=x+1-3sqrt((x-1)/(ax+1))
For which value of a ,(c) has no asymptote?
2. Relevant equations

3. The attempt at a solution
I know if a>0 then (c) will have 2 asymptote
And if a<o then (c) will have 1 vertical asymptote.
But I can't find any value of a so that (c) has no asymptote. Please give me some hints thank you.

2. Sep 18, 2015

### RUber

What is leftover after you have eliminated a>0 and a<0? There is only one value possible.

3. Sep 18, 2015

### Jeanclaud

a=0? But still it has an O.A.

4. Sep 18, 2015

### RUber

I don't see what you are saying? Your denominator would be (0x+1) = 1.

5. Sep 18, 2015

### RUber

I see now...you meant Oblique Asymptote.
I don't think this would have one.
By scale, yes -- the trend is in the direction of y = x as x gets large, but you can't argue that the function value gets closer to the line in the limit.

6. Sep 18, 2015

### Staff: Mentor

If a = 1, the part inside the radical approaches 1 for large values of |x|. That will give you an oblique asymptote.
The OP hasn't been very clear, but I think the asymptotes he's referring to are one that's oblique and one that's vertical.

7. Sep 18, 2015

### Jeanclaud

Then it will have only an asymptotic direction (parabolic branch). Thank you.

8. Sep 18, 2015

### SammyS

Staff Emeritus
Determine the domain of this function. That, of course, depends upon the value of a .