# Homework Help: Need help understanding limits

1. Feb 28, 2005

### shan

This isn't really a homework question so feel free to move is somewhere else. My teacher tried to explain this to me but I didn't understand it so I thought I'd try hearing another person's point of view.

The question is
lim n->infinity (2n-1)/(3n+1)

The working out I was showed is
= lim n->infinity (2n+2/3)/(3n+1) - (1 2/3)/(3n+1)
= 2/3 - 0 = 2/3

What I don't understand is how to work out that the lim n->infinity (2n+2/3)/(3n+1) is 2/3. We were shown rules for limits as n tends to infinity but I don't think I've come across the above situation before.

2. Feb 28, 2005

### jzq

$$\lim_{\substack{n\rightarrow \infty}}f(n)=\frac{2n-1}{3n+1}$$

Now, there's a shortcut. I don't know if you know it already but, if the powers are the same on the numerator and the denominator, just use the coefficients. Look at the 2n and the 3n. They have the same powers of 1 so use their coefficients 2 and 3. That will basically equal to 2/3. Hope that helps. Remember only when the powers are the same.

What you can also do is divide everything by the highest power in the denominator which is going to be n.

$$\lim_{\substack{n\rightarrow \infty}}f(n)=\frac {{\frac{2n}{n}}-{\frac{1}{n}}}{\frac{3n}{n}+{\frac{1}{n}}}$$

Cancel and you're left with:

$$\lim_{\substack{n\rightarrow \infty}}f(n)=\frac{{2}-\frac{1}{n}}{{3}+\frac{1}{n}}$$

Now plug zero into the n's and you're left with:

$$\lim_{\substack{n\rightarrow \infty}}f(n)=\frac{2}{3}$$

The method you were taught seems complicated to me.

Last edited: Feb 28, 2005
3. Mar 1, 2005

### shan

Cool, that makes it much clearer and simpler. Thanks very much :)

4. Mar 1, 2005

### Ouabache

I agree with jzq but with a small edition:
Since n approaches infinity in this question, I think jzq meant to say:

"Now plug zero in for (1/n)'s and you're left with:" ......

(that is because 1/infinity = 0).

A good online reference for "limit questions" can be viewed at:
http://www.jtaylor1142001.net/calcjat/Contents/CLimits.html

5. Mar 1, 2005

### Severian596

Man...limits are fun, aren't they?

6. Mar 8, 2005