I need to prove the following limit using epsilon delta method. I have come up this on my own just to practice skills.
$$\lim \limits_{x \to 1} \,(x^3 – 3x^2 + 2x) = 0 $$
So, I need to come up with some $\delta >0$ such that given $0 < \lvert x – 1 \rvert <\delta$, I need to prove that
$$ |(x^3 – 3x^2 + 2x) – 0| < \varepsilon \tag{1}$$
I can simplify eq $1$ as follows.
$$ |x| |x-1||x-2| < \varepsilon $$
When we have a non linear term, then we restrict $x$ to some distance from $1$. So, suppose we have
$ 0 < |x-1| < 10$. This leads to $-10 < x-1 < 10$. Doing some algebra, we can get inequalities
$$ 0 \leqslant |x| < 11 \tag{2}$$
$$ 0 \leqslant |x-2| < 11 \tag{3}$$
From eq 2 and 3, it follows that
$$ 0 \leqslant |x||x-2| < 121 $$
And since $0 < |x-1| $, it follows that
$$ |x||x-1||x-2| < 121 |x-1| \tag{4}$$
To show that $|x||x-1||x-2| < \varepsilon$ , its sufficient to show that $ 121 |x-1| < \varepsilon$. And this will happen if $ |x-1| < \frac{\varepsilon}{121} $. So, this is another restriction on $|x-1|$. So, we can let
$$\delta = \text{min}\left(10, \frac{\varepsilon}{121}\right) $$
So, we can let this be our choice of $\delta$. Its clear that $\delta > 0$. So, now the official proof will follow.
Let $\varepsilon > 0$ be some arbitrary real. Suppose $0 < |x-1| < \delta$. With the choice of $\delta$ we have done, this means that $ 0 < |x-1| < 10$. As demonstrated above, equation 4 follows from this inequality
$$ |x||x-1||x-2| < 121 |x-1| $$
But now, we also have $ |x-1| < \frac{\varepsilon}{121} $. It means that $ 121 |x-1| < \varepsilon $. Using this, we get that
$$ |x||x-1||x-2| < \varepsilon $$
which can be rewritten as
$$ | (x^3 – 3x^2 + 2x) – 0 | < \varepsilon $$
Since, $\varepsilon > 0$ was arbitrary to begin with, its proven that
$$\lim \limits_{x \to 1} \,(x^3 – 3x^2 + 2x) = 0 $$
Is the proof correct ?
Best Answer
This proof is fine, but manipulating the bounded part can sometimes prove more challenging.
Here it is easy since it is only $|x||x-2|$, but often you will get benefits while shifting the limit towards $0$ by setting $x=x_0+u$ with $u\to 0$.
The main benefit is that limits in zero are part of your mathematical toolbox, also $|u|<\delta$ is easy to manipulate compared to $|x-x_0|<\delta$.
For instance here $|x(x-1)(x-2)-0|=\overbrace{|u|}^{<\epsilon}\overbrace{|1-u^2|}^{\le 1}<\epsilon$
Granted we take $\delta=\min(1,\epsilon)$ so that $\begin{cases}|u|<\delta<\epsilon\\0\le u^2<\delta^2<1^2\implies|1-u^2|\le 1\end{cases}$
The inequalities did appear more naturally than when you calculated the $10$ and $\frac 1{121}$ factors.
Another advantage you will encounter in your epsilon-delta proofs is how it is easy to bound polynomials.
For instance imagine that in the problem instead of $|1-u^2|$ you had $|1-u^2+7u^3-2u^4|$.
Just proceed using the fact that for $|u|<1$ then $\cdots<|u|^4<|u|^3<|u|^2<|u|<1$.
The triangle inequality gives you $$|1-u^2+7u^3-2u^4|<|1|+|u^2|+7|u^3|+2|u^4|<1+1+7+2=11$$
Conclude by taking $\delta=\min(1,\frac{\epsilon}{11})$.