Proof of Limit: Lim (-3n^2 + n + 2)/(3n^2 -1)

[brickcitie]

Homework Statement

Find the following limit and prove your results using the definition of the limit:

Lim (-3n^2 + n + 2)/(3n^2 -1)

Homework Equations

The Attempt at a Solution

I passed the limit operator (if it is an operator) through and used it's properties to find that the limit is -1. As far as the proof I'm a little stuck. First I tried to solve the inequality which defines a limit for n in terms of epsilon, but I ran into trouble with that.

I think maybe one way to do it is to prove that it is a decreasing sequence with a lower bound of -1, although I'm not even sure if this is the case. I wish I knew how to format this so I could give a better idea of the question and my attempts at it, but I'm brand new and don't know how to do things as simple as write an equation using absolute values, limits, and epsilons. Any help at all is appreciated.

 

[Office_Shredder]

Try dividing the numerator and denominator by n²

 

[brickcitie]

I did that but and that's how I found that the limit is -1, but I'm still unable to complete the proof.

 

[Office_Shredder]

Once you do that, can you find the limit of the numerator and the denominator?

 

[brickcitie]

Yeah, I was able to find the limit of the numerator and denominator, and thus the limit of the entire expression, which is -1. I have no problem computing the limit, I am just having trouble proving that -1 is the limit using the definition of a limit, or the epsilon delta method.

 

[Office_Shredder]

I mean can you do the limit of the numerator and denominator using definition of a limit. Once you do that, an understanding of how the proof that the limit of a quotient is the quotient of two limits works should guide you the rest of the way.

Alternatively if you haven't seen that before, you should be able to just work off your intuition of how this is supposed to work. Why don't you show how far you're able to go and we can see how to progress from there

 

[brickcitie]

Ok awesome thanks for the help Shredder. I will hit the paper with this info and see if I can progress. Will post my progress soon.

 

[brickcitie]

Ok so I have made some slight progress I think, but I am unsure whether this is correct. Here is what I did:

Divided both top and bottom by n^2. Numerator is -3 + (1/n) + (2/n^2). Denominator is 3 - (1/n^2).

Went on to try and prove these using dfn of lim. Started with numerator because there was only a single expression containing n. I believe I was able to prove the denominator converges to 3 as long as n > 1/sqrt(epsilon). Is this correct? I hope so.

I am currently working on the numerator but I'm having trouble as I cannot solve for n in terms of epsilon because I have (1/n) + (2/n^2) in my inequality. Do I need to use the quadratic formula? This seems unlikely to me, and I can't imagine this is the most efficient way to proceed.

Any more tips shredder?

 

[brickcitie]

I know somebody can help me with this. I'm completely stuck.

 

[Office_Shredder]

In general you aren't going to be able to solve for n. All you want to do is hit the problem with a sledgehammer and hope that the piece that falls off is a good enough bound. In this example

$$\frac{1}{n}+\frac{2}{n^2}<\frac{1}{n}+\frac{2}{n}$$ (you should figure out why this is true)

So if you can find a bound for n for the right side, you find one for the left side as well (again, make sure you understand why)

 

[brickcitie]

Ok I think I see what you mean here. So I need to find a bound for an expression that is easier to solve for n, but is greater than my expression.

Gonna attempt to solve with this method. This freaking problem sucks.