I'm asked to prove, using the definition of convergence, that limits approach a certain value.
For example, $$\lim_{n\rightarrow\infty}\dfrac{n^2+4}{n^2}.$$ I can see that it converges to $1$, but I'm not sure how to go about the proof of it using the definition of convergence. (There exists an $N$ such that for all $n\geq N$, $\mid x_n-x\mid<\epsilon$ for all $\epsilon>0$ )
Note that
$$\left|\frac{n^2 + 4}{n^2} - 1\right| = \left|\frac{4}{n^2}\right| = \frac{4}{n^2}\tag{*}$$
Given $\epsilon > 0$, the rightmost side of $(*)$ is less than $\epsilon$ when $n$ is greater than $\ldots$