<s>[INST] Proof using formal definition: Infinite limit I was wondering how get the proof of this limit:
$$\lim\limits_{x\to -\infty}\dfrac{{x^2} - x + 1}{x + 4} = -\infty$$
The problem is that I don't know what to do for find the appropriated values to make valid the implication of the formal definition (epsilon-delta).
I would appreciate if somebody can help me. 
 [/INST] Note that
$$\frac{x^2-x+1}{x+4}=\frac{(x+4)(x-5)+21}{x+4}.$$
You need to show that given $M<0$, there exists $N<0$ such that $\frac{x^2-x+1}{x+4}<M$ for all $x<N$. Let $N=\min\{M,-4\}.$ Then if $x<N$,
\begin{align*}
\frac{x^2-x+1}{x+4}&=\frac{(x+4)(x-5)+21}{x+4}\\
&=(x-5) + \frac{21}{x+4}\\
&<(N-5) + \frac{21}{x+4}\\
&<M+\frac{21}{x+4}\\
&<M.
\end{align*}
We can drop the $\frac{21}{x+4}$ because $x<N \leq -4$, so $x+4$ is negative, so $\frac{21}{x+4}<0$.
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