Find the limit of the sequence n^3/(2n^3 + n)

[walter9459]

Homework Statement

Find the limit for the following sequence and then use the definition of limit to justify your result.

Homework Equations

n^3/(2*n^3 + n)

The Attempt at a Solution

I found the limit as n --> infinity is 1/2. I think the next step is to set up the equation as follows:

n^3/(2*n^3 + n) = 1/2

But then I'm not sure if I should add n+1 to each side.

 

[Dick]

Don't do any such thing. Look at the definition of a limit. Divide numerator and denominator of your expression by n^3. So you have 1/(2+1/n^2). Now you want to find an N for every epsilon such that n>N implies |1/(2+1/n^2)-1/2|<epsilon for any epsilon however small. Isn't that what the definition of limit said? Can you do that?

 

[walter9459]

okay, I came up with 1/(2+1/n^2) which is where I came up the idea that the limit as n---->infinity is 1/2. But then if I set it to be < epsilon. How do you know what epsilon should be? I am really struggling with this concept. I appreciate your help and patience.

 

[Dick]

Think of epsilon as a really small positive number. You want to find an N so large that 1/2-1/(2+1/n^2)<epsilon for n>N. If you do the algebra that's 1/(4N^2+2)<epsilon. Since 1/(4N^2+2)<1/(4N^2) if 1/(4N^2)<epsilon then you have a good N. I would pick an N bigger than 1/(2*sqrt(epsilon)). Do you see why? If you can find an N for every epsilon then the limit is valid.