Limit of a sequence and my idea $$a_n=\frac{n^3-7}{n^4+3}$$
So, I am supposed to prove that the sequence converges to $0$...I am kind of stuck, couldn't find any $N$ that fits...Can someone help?
Small idea:
Since the sequence obviously converges to $0$.
Given $\epsilon >0,|\frac{n^3-7}{n^4+3}-0|<\epsilon$, Since $\frac{n^3-7}{n^4+3}<\frac{n^3}{n^4}$ for any $n\ge 2$ we can say $\frac{n^3}{n^4}=\frac{1}{n}<\epsilon$ so we have $\frac{n^3-7}{n^4+3}<\frac{n^3}{n^4}<\epsilon$ Since i didn't consider the case where n is negative, i dont know if this is correct...
 A: 
Since $\frac{n^3-7}{n^4+3}<\frac{n^3}{n^4}$

How do you know this is true? I mean, it is, but some justification would be nice.

Other than that, your proof is OK, so long as you mention that $$\frac{n^3-7}{n^4+3}>0$$ somewhere in your proof.
That way, you can claim that if $\frac{1}{n}<\epsilon$, you have
$$\left|\frac{n^3-7}{n^4+3}\right|=\frac{n^3-7}{n^4+3}<\frac1n<\epsilon$$
and you can more or less conclude the proof (depending on how strict you need to be, you may need to write a couple more lines to go to the letter of the definition though).
If you need to be very strict, then you need to prove the exact definition of a limit, and then show that that definition is satisfied. Remember, the definition says that $L$ is the limit of $a_n$ if and only if, for every $\epsilon > 0$, there exists some $N$ such that if $n>N$, then $|a_n-L|<\epsilon$.  So, for a full formal proof, your proof needs to look something like this (where you fill in the blanks, of course):

*

*Let $\epsilon > 0$.

*Then, let $N$ be such that ____________. (here, you must select the correct value of $N$)

*Let $n>N$. Then, $$\left|a_n-0\right| = \cdots < \cdots < \epsilon$$ which means that, by definition, $0$ is the limit of the sequence $a_n$. (here, if you chose the correct $N$, you should be able to prove that $|a_n-0|<\epsilon$).


You do not need to consider the case when $n$ is negative if you are looking at $$\lim_{n\to\infty} a_n,$$
because you are only interested in what happens for large values of $n$.
