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I am having trouble understanding part of the solution to this simple problem.

$\lim_{x \to 2} (x^2 + 3x) = 10$

Solution:

Let $\epsilon > 0$

$| x - 2 | < \delta$ and $| x^2 +3x -10 | < \epsilon$

since $x^2 +3x -10 = (x - 2)^2 + 7x -14 = (x - 2)^2 + 7x -14 = ( x -2 )^2 +7(x-2)$

$|(x-2)^2 +7(x-2)| \leq |(x-2)|^2 +7|(x-2)|$

$\delta^2 + 7\delta < \epsilon$

let $\delta$ be the minimum of $1$ and $\epsilon/8$, $\delta^2 \leq \delta$.

then $8\delta < \epsilon$

$\delta < \epsilon/8$.

My Question:

I worked my way through the question down to $\delta^2 + 7\delta < \epsilon$

I then got confused by the end of this statement

Let $\delta$ be the minimum of $1$ and $\epsilon/8$, $\delta^2 \leq \delta$.

and in particular $\delta^2 \leq \delta$. I see how this allows me to prove the limit but I cannot make sense out of $\delta^2 \leq \delta$.

Could anyone explain this to me?

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    @MWright: It wasn't me who beat you to it. You can see who edited a post by clicking on the time where it says "edited 2012-12-26

2 Answers 2

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By choosing $\delta$ the minimum of $1,\frac{\epsilon}{8}$ no matter what the value of $\epsilon$:

$\delta \leq1 \Rightarrow \delta^2 \leq \delta \Rightarrow \delta^2+7\delta\leq 8\delta$

3

We have been challenged with an $\epsilon$, perhaps $\epsilon=1/1000$. We want to come up with a $\delta$ such that if $|x-2|\lt \delta$, then for sure $|x^2+3x-10|\lt \epsilon$.

Suppose that after some calculation, we announce that $\delta=\epsilon/8$ does the job. Then triumphantly the challenger could say that she had $\epsilon=47$ in mind, and that in that case taking $\delta=47/8$ is insufficient. Of course, that is not playing fair. But we might as well come up with a $\delta=\delta(\epsilon)$ that always works.

Some algebra shows that $x^2+3x-10=(x-2)^2+7(x-2)$. So we want to make sure that $|(x-2)^2+7(x-2)|\lt \epsilon$. Note that $|(x-2)^2+7(x-2)|\le (x-2)^2+7|x-2|.$ We want to make the right-hand side "small," by choosing $x$ appropriately close to $2$. Suppose we had been given a ridiculous $\epsilon$, like $47$. If $x$ is within $47$ of $2$, the number $(x-2)^2$ could be very large. So the first task is to make sure $\delta$ is small enough not to allow $(x-2)^2$ to be large.

So we say first of all, let $\delta\le 1$. Then if $|x-2|\lt \delta$, it follows that $(x-2)^2\lt \delta$. For $(x-2)^2=(x-2)(x-2)$. The "first" $x-2$ has absolute value $\le 1$, and the second has absolute value $|x-2|$, so the product has absolute value $\le |x-2|$.

It follows that as long as $\delta\le 1$, we have $(x-2)^2+7|x-2|\le 8|x-2|$. To make sure this is $\lt \epsilon$, it is enough to make $\delta=\frac{1}{8}\epsilon$.

However, in deriving our simplified inequality, we assumed that $\delta\le 1$. So we know that everything will work if $\delta=\min(1,\epsilon/8)$.

It might have been better to observe that $(x-2)^2+7|x-2|=|x-2|\left(|x-2|+7\right).$ Then it is clear that if $|x-2|\le 1$, then $(x-2)^2+7|x-2|\le 8|x-2|$.