Example
Prove that \(\lim\limits_{n \rightarrow \infty} \frac{2n^2 + 1}{n^2 - n - 10} = 2\).
Strategy
We want to choose \(N\) such that we get this inequality (so we solve for \(n\))
$$
\begin{align*}
\left| \frac{2n^2 + 1}{n^2 - n - 10} - 2 \right| &< \epsilon \\
\left| \frac{2n^2 + 1 - 2(n^2 - n - 10)}{n^2 - n - 10} \right| &< \epsilon \\
\left| \frac{2n^2 + 1 - 2n^2 + 2n + 20}{n^2 - n - 10} \right| &< \epsilon \\
\left| \frac{2n + 21}{n^2 - n - 10} \right| &< \epsilon \\
\frac{2n + 21}{n^2 - n - 10} &< \epsilon \quad \text{(when $n \geq 10$)}.
\end{align*}
$$
At this point, we want to bound this term but we can’t just remove the \(-10\) or \(-n\) since this will make the denominator bigger and thus the fraction smaller while instead we want the fraction to stay the same or get bigger. Since if the fraction gets smaller, then we can’t conclude that are original fraction is less than \(\epsilon\). It might be bigger! To guarantee that the original fraction is less than \(\epsilon\), we instead want to find a term that’s larger. We can then bound the larger term and hence our original fraction will also be less than \(\epsilon\) since the bigger one is! Hence observe that
$$
\begin{align*}
n < \frac{1}{2}n^2 \quad \text{if } n > 2
\end{align*}
$$
Also observe that
$$
\begin{align*}
10 < \frac{1}{3}n^2 \quad \text{if } n > 6
\end{align*}
$$
Then
$$
\begin{align*}
n^2 - n - 10 &> n^2 - \frac{1}{2}n^2 - \frac{1}{3}n^2 \quad \text{(if $n > 6$)} \\
&= \frac{1}{6}n^2
\end{align*}
$$
So now since \(n^2 - n - 10 > \frac{1}{6}n^2\), we can replace the whole denominator with the new simpler and smaller term \(\frac{1}{6}n^2\). This gives us a larger fraction so the bound isn’t as tight anymore but it makes things simpler
$$
\begin{align*}
\frac{2n + 21}{n^2 - n - 10} &< \epsilon \\
\frac{2n + 21}{\frac{1}{6}n^2} &< \epsilon \\
\frac{6(2n + 21)}{n^2} &< \epsilon \\
\frac{12n + 126}{n^2} &< \epsilon \\
\frac{12n + 126n}{n^2} &< \epsilon \quad \text{($126n > 126$)}\\
\frac{138}{n} &< \epsilon \\
\frac{138}{\epsilon} &< n.
\end{align*}
$$
Formal Proof
[TODO]
References