Limit existence and infinity

One if the awesome things about studying on my own, is the flexibility of choosing my own schedule and resources. The drawback, as I have mentioned before, is that different resources have different points, methods and even explanations. If your resources also vary in timeframe, those issues become more apparent. Let's take for example the limits of 1/x as x approaches 0 from the negative and from the positive side.

If we look at a graph of 1/x we can see that as x approaches 0 from the negative side, the graph starts to approach the y axis but not touch it. Since x cannot be 0 (1/0 is undefined), then the y axis, where x=0, is an asymptote, a line that will never be touched by the graph. That means that our graph will get closer and closer to the y axis for all infinity without touching it. Therefore the limit of 1/x as x approaches 0 from the negative side, in the strictest definition of a limit, does not exist and that is how my text book defines it.

My online class however, also defines this limit as negative infinity and as positive infinity when you approach 0 from the positive side. The Profesor explained that it was a more informative way to describe the limit of 1/x as x approaches 0.

My problem came along when I did my textbook homework after my online lesson. While I was checking my answers I noticed I had gotten two incorrect ones. They were the limits described above. In the textbook the right answer was that the limits did not exist, I had answered that they were positive and negative Infiniti. Being a student on my own, I had to retrace my steps, watched my class again and do a lot of research to find out if I had gotten the right answer. When I reviewed my online lesson I heard the explanation the professor made about these limits not existing and how using infinity gave us more info about what was happening.

If I had not been able to re-watch my class, I would probably still be looking for the answer. In a traditional classroom my doubt would have been put to rest in a second by the teacher. On my own I had to figure it out myself. I see advantages and disadvantages to both situations.

How about you?

 

On limits and life

These last two weeks I've been studying limits in various ways, from various sources, and different media. I've used videos, sites, ebooks, paper books, ad apps. It is great to see how each reference approaches the topic differently, and how the central concepts emerge from the gathering of information.

I am starting to see why many people regard Calculus as beautiful. Being able to find the equation of a line that is tangent to a curve using limits is breathtaking. I am not kidding. There was something awe inspiring when out of some algebra and some elementary Calculus I arrived at a formula, that when graphed, touched a curve at exactly one point before continuing in its path. Can you see the significance of that? These two functions for an instant, touched, and then moved on.

To delve deeper into this existential stream of consciousness about Calculus, consider that a limit tells us about where a function is heading. It doesn't care about what happens when the function gets there. It cares more about what happens as it gets closer and closer to a given point. Therefore, in Calculus as in life, the most important thing is the journey rather than the destination.That gets me to think about the people we meet briefly once in our lives. Those chance encounters might not change our paths, but there is a cosmic record that they happened. At that moment when two people meet briefly, you can describe each of them as being together. In other words we could say that at 9:00pm on Sunday, May 19th, 2013, John met Mary and Mary met John. Therefore John and Mary became part, in that instant, of the greater formula of life as variables sharing the same time and space coordinates. And that point in time, much like a limit, gives us information about where each of them is and how they are behaving.

I am liking this calculus stuff more and more each day.

Let me know your thoughts.

 

Slight Detour: The story of a cube

As I was gathering resources and reference for Calculus. I ran by a page (that I cannot locate now), of a math tutor that made a small comment about how he also had a Rubik's cube solving page.

 

Flash back at least 11 years ago.

 

For a holiday (I can't remember which) my girlfriend of at least 8 years years (who is now my wife of 10 years) gave me a picture cube as a gift.

After briefly looking at all the pictures of our life together, in fast, deliberate motions, I scrambled the cube. The look of horror on my future's wife face is still etched in my mind.

 

I looked down at the cube and shared her concern. I had never solved a Rubik's Cube. We knew no one who had, therefore this cube of our pictures would never be rearranged again.

 

I gave it all I had, for hours and days and weeks I tried to solve it. But I could not. The cube sat in my car for at least two years. One day, after we got married, my wife found it and gave it a sad forlorn look. She took it and stored it with the rest of our pictures and heirlooms. I felt like the worst person on earth for being foolish enough to scramble that picture cube.

 

Flash forward to two weeks ago.

 

When I saw that page on how to solve a Rubik's Cube using algorithms. I knew what I had to do.

 

As fate had it, one day we were shopping at Party City and they were selling mini Rubik's cube for 89 cents. I grabbed one as casually as I could and bought it.

 

Like I said, I never found the first page I used, but afterwards I got these others and these were the ones that helped me the most. Beginner's Solution to the Rubik's Cube. Beginner's Rubik's Cube Solutions. In the beginning, as I read these I felt overwhelmed, almost as overwhelmed as I felt doing Calculus without precalculus back in January. They were talking about Faces, primes, clockwise and counter-clockwise moves. They also mentiones middle pieces, corner pieces, edge pieces. Then they would give me string of algorithms that looked like this: R2 U F B' R2 F' B U R2.

 

There was no way this could work, could it? My first attempt with my mini cube, after at least 3 hours, ended in utter failure. My wife, who thought I had gotten my self another hobby besides calculus was not impressed. I had kept my real intentions secret from her. And on top of everything, she thought I was just cheating using formulas to solve a cube. At the time, so did I. I felt like I was painting by numbers. All I had to do was follow those instructions to the letter and I would solve a cube.

 

By the time I had solved my mini cube, I felt otherwise. This was not painting by numbers. Solving a Rubik's cube with the beginner method was about recognizing patterns and executing moves to make the pieces go where you wanted them to...without messing with the other pieces you had already done.

 

After solving my mini cube, I did not feel ready enough to tackle our picture cube. So I looked for more practice. I got the Rubik's cube app for my iPhone and Ipad and bought a "magic cube" from China on EBay. My goal was to practice with these as much as I could.

 

As I waited for my Chinese cube to arrive, I practiced with the app. I found it hard to use at first because the controls were weird. But after a while, I was "fluent" in it.

Still remembering the combination of moves that I needed to make and when I was supposed to make them still took me a long time.

 

My first time solving a cube in the app took me 1 hour 49 minutes. And I had to refer back and forward to my notes. More than once my wife found scraps of paper with algorithms on them...I got some weird looks from her.

 

The great thing about having the app is that I could fire it up at a moments notice if I was waiting in line, in an elevator, or in my lunch break. I could even pause the game and pick it up later at night right before bed. I was getting good practice out of this. After 5 tries I could keep my time under an hour. After 10 tries I could solve it a shade under 15 minutes. In the last 5 tries I could solve a cube in about 6 minutes.

 

I now felt ready to solve our picture cube.

 

All last week I have been asking my wife if she remembered were we had put that picture cube. I still did not tell her what I wanted to do, but I knew she knew. What I bet she did not know was that I could solve it this time. After a few attempts at finding it, and making a mess of our closet, I got my hands on it. The cube felt weirdly solid in my hand. As if it had gathered the mass of 10+ years just waiting for this moment.

 

I looked at it and remembered how hard I had tried to solve it the last time. Yet, all that was solved was a single face. It showed a picture of a kiss I gave my future wife after I had caught the garter from my brother-in-laws wedding. I would propose to María soon after that. It seemed fitting that I would attempt solve this cube on the year of our 10th wedding anniversary.

 

However, after I look a the rest of the pictures, I panicked. I could not tell which pieces belonged together. Many of the pictures had parts that were almost the same color. This would not be as easy as I had hoped. María was walking around the house doing some stuff and would check in me from time to time. When she did I asked her questions that must not have been reassuring to her like: where do you think this piece goes? She would give me her best guess and walk on.

 

After about 20 minutes it all clicked and I was back on track. A minute later I had solved all the pieces. However, I was not done. This is a picture cube which has a key difference than a regular cube.

In a picture cube the center pieces need to be rotated to fit the rest of the pattern. If you don't, you will have a solved cube looking like my mini cube did: I bet Jessie would love to have her face back in its proper orientation. Before you say anything...yes, my mini cube was of Toy Story...moving on!

Having centers not rotated properly can be quite distressing to many people. Because you can actually see it happening while you are solving the cube. If you type, Rotate center pieces, in google, you will get dozens of hits on this problem and how to solve it. The one I found to help me most was this video. The process involved a very simple algorithm repeated 5 times to turn a center piece 90 degrees clockwise and another center piece counterclockwise.

To finish my cube I would have to do the algorithm twice. Since I had to do 4 moves to complete the instructions once. That meant I had to do 40 moves to rotate all center pieces to their proper place. If I make a single mistake, I could scramble the cube in such a way that all my effort would be wasted. And guess what, at move 38, I lost track of my next move. I had no idea what I had to do next and the cube looked horribly scrambled.

I was heartbroken. I knew I was really close, but if I made the wrong move It was over. I just sat there, looking at all the faces of the cube and see if I could get back on track. A week ago I would have been lost, but after all that practice, I could see the relationships between pieces clearer. I felt a rush of adrenaline as my brain registered that I had already solved the cube. It must be the same rush a chess player gets when he or she knows the game has already been won a few moves ahead. I took a deep breath and mave the last two turns.

And there it was...solved. After all those years.

I showed it to María and the look on her face was amazing. I sat her down and told her how this cube had been on my mind for many years. how I always wanted to solve it. To rearrange all those pictures of our lives together. It was symbolic for me. And then she told me something that melted my heart.

A Rubik's cube has 54 squares. 9 squares per face per 6 faces. María told me that When our picture cube arrived the first time her hands. She was disappointed. The sticker that had the pictures looked really fragile to her. She knew that unless she did something, it would deteriorate and fade really soon. So she cut 54 squares of clear adhesive paper and covered all of them, one by one. It must have taken her hours of painstaking and risky work to get that cube to me. No wonder she was so chocked when I just scambled it the first time, and so sad when she found it still unsolved.

Now that we had it back we put it in a place of honor. Our bedroom table. To be looked upon by us and our little girl for many years to come.

 

Pedagogical Quandary

It's been a while since I last wrote. I have been busy studying but not writing about it. There's no good reason for me not writing, even when I have made little progress. I guess if I wrote about what I've been doing it would be repetitive.

 

I am starting Calculus on my own with no formal teacher or course. I am relying on three textbooks and video lessons. Still after starting Calculus a couple of weeks ago, I have not gotten that far because all three books start with pre-calculus. The only resource I have that starts with Calculus are the video lessons but they don't give me any homework exercises to practice.

 

Therefore here is my quandary: should I skip the precalculus and go right to calculus since I have 10 weeks worth precalc? Or should I glance over what the books have to say about precalculus just in case they shed light into how they will cover calculus?

 

I have been inclined to do the former and check out the preparation chapters. That means that I have studied the precursors of limits (slopes of secant lines) at least in three different ways. At least, each time I get the introduction to limits I understand them better.

 

My other quandary is with practice exercises, I want to get as much homework as I can but all my books only have the answers for the odd numbered items. That took me back to school. I remember being assigned odd numbered items at home for practice and even numbered items for hand in assignments. It always made me anxious when I could not check wether or not I had done the exercises well.

 

In high school, Mr. Quintero, our notorious but brilliant math teacher changed all that. He had no problem assigning odd numbered exercises to hand in. He realized the back of the book only gave us the answers, so he would put all the weight in the process. That way you could either get the exercise right and check the answer, or get it wrong and work through it to find out what happened. I found this approach far more effective and instructive.

 

Without a teacher's help, all I have is the answers provided by the book to know if I am right. I guess that is a trade off I'll have to work with.

 

What do you think?