proving limits exist Prove $\lim\frac{ 4n^3+3n}{n^3-6}= 4$.
I basically need to determine how large $n$ has to be to imply
$$\frac{3n+24}{n^3-6}<\epsilon$$
the idea is to upper bound the numerator and lower bound the denominator.
For example, since $3n + 24 ≤ 27n$, it suffices for us to get $\frac{27n}{n3-6} < ε$.
it is inferred, thereafter, that all we need is $\frac{n^3}2 ≥6$ or $n^3 ≥12$ or $n>2$.
this is where I have a problem. I don't understand how we got to the point where all we need is  $\frac{n^3}2≥6$.
 A: If $\frac{n^3}2\ge 6$ then
$$n^3-6\ge n^3-\frac{n^3}2=\frac{n^3}2 $$
is your desired monomial lower bound for the denominator.

In fact, we are allowed to be wasteful and may begin right away with, say, assuming $n>1000$.
That makes $3n+24<3.024 n$ and $n^3-6>0.999999994n^3$ and so the error term certainly $<\frac{3.025}{n^2}$. But that doesn't matter for the sole purpose of proving the limit where any other bound of the form $<\frac C{n^2}$ (or even weaker) is good enough.
A: 
I basically need to determine how large $n$ has to be to imply

$$\frac{3n+24}{n^3-6}<\epsilon$$

the idea is to upper bound the numerator and lower bound the denominator.

For $n > 9$, $0 < (n^3 - 64n),$ which is a lower bound for the denominator.  
For $n > 9$, $(n-8) > 1 \implies$ that the numerator is 
less than $3(n+8)(n-8) = 3(n^2 - 64).$
Therefore, the fraction is less than
$$\frac{3(n^2 - 64)}{n^3 - 64n} = \frac{3}{n}.$$
Therefore, choose $n$ such that $n > 9$ and 
$\frac{3}{n} < \epsilon \implies \frac{3}{\epsilon} < n.$
A: The most dominant term in numerator is $4n^3$ and the most dominant term in denominator is $n^3$ so the limit is $4$.
A: Just look at the highest power of $n$ upside and downside (and divide everything by that power of $n$):
$$\lim_{n\to\infty}\frac{4n^3+3n}{n^3-6}= \lim_{n\to\infty}\frac{4+\frac{3}{n^2}}{1-\frac{6}{n^3}}=\frac{4+\frac{3}{\infty}}{1-\frac{6}{\infty}}=\frac{4+0}{1-0}=\boxed{4}$$
A: For $n \neq 0$,$$ \frac{ 4n^3+3n}{n^3-6}=  \frac{ 4+\frac{3}{n^2}}{1-\frac{6}{n^3}}.$$
You should be using the properties of the limits of sequences, like here, or Rudin's PMA Theorem $3.3$. You should prove the properties first, then use them. This then reduces your epsilon-delta task significantly for questions like this one.
