# Limits and Vertical Asymptotes

1. Nov 6, 2012

### knowLittle

1. The problem statement, all variables and given/known data
I found that the domain of f was defined for all reals x<-1/3 and x>1/3. Now, I need to find the limits and vertical asymptotes of
$\lim _{\chi \rightarrow \dfrac {1} {3}}\dfrac {x+2} {\sqrt {9x^{2}-1}}$

According to Wolfram Alpha, there is no limit. But, I found that the limit approaches 54(80/9)^(1/2).

3. The attempt at a solution
$\dfrac {\dfrac {x} {x}+\dfrac {2} {x}} {\dfrac {\sqrt {9x^{2}-1}} {\sqrt {x^{2}}}}=\dfrac {1+\dfrac {2} {x}} {\sqrt {9-\dfrac {1} {x^{2}}}}$
Then, I use L' Hospital:

$\dfrac {-2x^{-2}} {\dfrac {1} {2}\left( 9-x^{2}\right) ^{-1 / 2}\left( -2x\right) }=\dfrac {2} {x^{3}}\sqrt {9-x^{2}}$

So, the limit approaches 54 *(80/9)^1/2

Last edited: Nov 6, 2012
2. Nov 6, 2012

### SammyS

Staff Emeritus
L'Hôpital's rule can't be applied here.

This limit is not of the form 0/0 or ∞/∞ .

3. Nov 6, 2012

### knowLittle

You are right, but how can I proceed?

4. Nov 6, 2012

### Zondrina

Try multiplying by 1.

5. Nov 6, 2012

### knowLittle

I am trying the squeeze theorem now, but still no answer. There have to be some sort of crafty manipulation to the equation. It's very easy to find the limit at infinity, though.

6. Nov 6, 2012

### Zondrina

Hmm now that I took a closer look at this. You should get no limit at all. Think about what happens as you approach 1/3 from the right and 1/3 from the left.

7. Nov 6, 2012

### knowLittle

I know that the function grows without bound as x approaches x=-1/3 and x=1/3, but I need to prove it.

8. Nov 6, 2012

### Staff: Mentor

Easy, but not relevant.

Since the domain is (-∞, -1/3) U (1/3, ∞) it makes no sense to approach 1/3 from the left. The function is not defined on [-1/3, 1/3].

The problem should be restated as this:
$$\lim_{x \to 1/3^+}\frac{x + 2}{\sqrt{9x^2 - 1}}$$

The limit can be evaluated directly by noting what the numerator is doing and what the denominator is doing as x gets close to 1/3 from the right.

9. Nov 6, 2012

### knowLittle

I see that the numerator is approaching 2.333... and the denominator approaches 0.
I know that, when lim x-> 0 1/(x^2) it's very clear that the function grows without bounds, but in this occasion I just can't see how it grows.

10. Nov 6, 2012

### SammyS

Staff Emeritus
Yes, the denominator approaches 0, but from what direction?

Is the denominator ever negative?

11. Nov 6, 2012

### Staff: Mentor

The answer to Sammy's question is crucial to determining what the limit is.

12. Nov 6, 2012

### knowLittle

The denominator is never negative. It approaches 0 from the right hand side.

Is this the same as saying: (any constant)/ (a number infinitely small that never really reaches zero) ?
Then, it follows by the same principle as 1/ (x^2) as x->0 grows without bound?

13. Nov 6, 2012

### Staff: Mentor

I think you have the idea, but you're not saying it very well.

If the numerator is a positive constant, and the denominator approaches zero but remains constant, then the right-hand limit of this quotient is infinity.

14. Nov 7, 2012

### knowLittle

What do you mean by, "the denominator approaches zero, but remains constant"?
How do you know that it remains constant? How so?

15. Nov 7, 2012

### SammyS

Staff Emeritus
I think Mark meant to say
the denominator approaches zero, but remains positive.​

16. Nov 7, 2012

### knowLittle

Thank you very much to all. Now, it remains clear that the same principle of limit is there.

Thank you, again!