I have a question regarding the more complicated proof in example 3 in the attachment. When you see that you can't define δ = ε/│x+5│and replace │x+5│ with M, why must│x+5│≤ M. Could you please explain this? Thanks so much for this post!
I know that direct and indirect geometric proofs using the statement vs reason chart are not taught in most high schools across the USA. They cover the basics of proving two triangles are congruent but nothing beyond simple triangles. Teachers are forced to test prep their students. Teachers need to teach and spend less time preparing students for standardized exams. We need more education and less test prep.
I have a question regarding the more complicated proof in example 3 in the attachment. When you see that you can't define δ = ε/│x+5│and replace │x+5│ with M, why must│x+5│≤ M. Could you please explain this? Thanks so much for this post!
It's explained in the very next line. Basically, if x is reasonably close to 5, then we want a function of $\displaystyle \begin{align*} \epsilon \end{align*}$ that ends up being an upper bound for $\displaystyle \begin{align*} |x - 5| \end{align*}$. So say we want x to be 1 unit away from 5, then that means we have
$\displaystyle \begin{align*} |x - 5| &< 1 \\ -1 < x - 5 &< 1 \\ 4 < x &< 6 \\ 9 < x + 5 &< 11 \\ | x + 5 | &< 11 \end{align*}$
So that means as long as $\displaystyle \begin{align*} |x - 5| < 1 \end{align*}$ that means $\displaystyle \begin{align*} |x - 5| < \frac{\epsilon}{11} \end{align*}$, and thus an appropriate choice for $\displaystyle \begin{align*} \delta \end{align*}$ is $\displaystyle \begin{align*} \min \left\{ 1, \frac{\epsilon}{11} \right\} \end{align*}$.
Ok, I'm just trying to fully grasp these concepts. What do you mean when you say a function of ε that ends up being an upper bound for │x-5│? I don't think I'm seeing the bigger picture here.
That's the whole point of an $\displaystyle \begin{align*} \epsilon - \delta \end{align*}$ proof. To prove that $\displaystyle \begin{align*} \lim_{x \to c} f(x) = L \end{align*}$ you need to show that for all $\displaystyle \begin{align*} \epsilon > 0 \end{align*}$ there exists a $\displaystyle \begin{align*} \delta \end{align*}$ which is a function of $\displaystyle \begin{align*} \epsilon \end{align*}$ such that $\displaystyle \begin{align*} 0 < |x - c | < \delta \implies \left| f(x) - L \right| < \epsilon \end{align*}$.
Loosely speaking, it's saying that when you set a tolerance for $\displaystyle \begin{align*} f(x) \end{align*}$ which we call $\displaystyle \begin{align*} \epsilon \end{align*}$, as long as you are sufficiently close to $\displaystyle \begin{align*} x = c \end{align*}$, then you are ensured that the distance between $\displaystyle \begin{align*} f(x) \end{align*}$ and the limiting value $\displaystyle \begin{align*} L \end{align*}$ is always less than your tolerance.
So this "upper bound for |x - 5|" is the $\displaystyle \begin{align*} \delta \end{align*}$ function you are looking for in this case.
I am on my IPad so going back and forth is almost impossible, but I think there are two issues that you are missing. First, if the absolute value is less than one fifth, it is necessarily less than one third. So one fifth works, but it is unnecessarily restrictive. Second and more important, 0 < a < b < c is equivalent to 0 < 1/c < 1/b < 1/a. As I say, it is hard for me to figure out what you are thinking and how it relates to an attachment on a different page without being on my PC, but as far as I can tell under not good conditions, there is no error in the example.
If you explain what you think the error is, I'll look again tomorrow on the PC.
I feel like the direction is shifting from implanting logic and reasoning in students to forcing a desired output regardless of method. Basically the curriculum is shifting so that students have to find other resources to fully understand what they are actually covering in class, but are given just the basic tools to be able to take an any question given by the school which are generally rigid in format (input) and produce the desired answer (output). . . proofs don't seem to be part of the equation in my experience. Thus, this is really good information for anyone who wants to really understand what a limit is beyond the mathless math classes they are taking.
I am currently volunteering at the local library by tutoring high school students in math. So I am looking at mathematical education from a distance and may be grossly in error. But I do not like what I see.
The concept of proof IS short changed. This is probably the easiest defect to excuse, but a very hard one to correct. The reason I suspect is that over the last 200 hundred years the standards for proofs among professional mathematicians have gone up drastically, but kids in high school do not see the subtle problems being addressed by modern rigor. The solution adopted seems to be to just skip proof entirely.
Another problem is that concepts are introduced and then not used for months or even years. I'll give an example that came up today. I have a bright student who is being introduced to the exponential and logarithmic functions. This is in chapter 6. In chapter 2, there was a review of functions, which was introduced two years ago and then never really used intensively. These delays lead kids to decide that math is a whole mess of definitions and concepts that are not even useful within math. We spent half the hour going over functions, particularly the ideas of inverse function and of functions being a rule or relation that ties the numbers in one set to the numbers in another set in an unambiguous way.
And despite the fact that the text being used in the local public school system is explicitly tied to using a specific model of graphing calculator and clearly recognizes how much of the drudgery in the purely mechanical aspect of applying math can now be avoided, at least 80% of the exercises are purely mechanical rather than word problems that require intuition and the ability to apply mathematical concepts and techniques to apparently non-mathematical material.
I get very depressed.