simple question about epsilon and estimation turing machines
Suppose there exists $k&gt;0$ such that $\lim_{n\to\infty} \left(3-\frac7n\right)=3-k$. 

By the definition of a limit,
$$\lim_{n\to\infty} f(n)=L\Leftrightarrow \forall\epsilon&gt;0,\exists N&gt;0,n&gt;N\implies |f(n)-L|&lt;\epsilon$$

Here, we need to check:
$$\lim_{n\to\infty} \left(3-\frac7n\right)=3-k\Leftrightarrow \forall\epsilon&gt;0,\exists N&gt;0,n&gt;N\implies \left|k-\frac7n\right|&lt;\epsilon$$

Now, disproving this just requires a suitable choice of $\epsilon$. Consider $\epsilon=\frac k2$. We have for all $n&gt;\frac{14}{k}$, $\left|k-\frac7n\right|&gt;\frac k2$. So, there does not exist any $N&gt;0$ satisfying the condition, and as  a result, no such $k&gt;0$ can exist.