I was happy with the amount of progress I made this week. I’m now 75% of the way through the unit Limits and Continuity with only have handful of videos and exercises, as well as the unit test to get through so I think there’s a strong chance I’ll finish the unit by next week. It felt like I did a ton of work this week but, surprisingly, just realized I only wrote out a few pages worth of notes. In total I watched 27 videos this week which would account for me not writing as many notes as I typically do in weeks where I work through more exercises. On the bright side it means this post will be a bit shorter than usual. 🤷🏻‍♂️

The first thing I learned this week is what’s known as the Squeeze Theorem. My layman’s explanation for this theorem is, if there are three functions, f(x), g(x), and h(x), and you know that the limit of f(x) ≤ g(x) ≤h(x) but you don’t know the limit of g(x) but you do know that the limit of f(x) = L = h(x) then you can then conclude that g(x) = L, as well.

My first reaction when learning this theorem was, “…well duuuh”, since it seemed so self evident. I didn’t see the purpose or value in learning it but then Sal used this theorem to conclude that lim_x->0(sin(x)/x) = 1 which is counterintuitive since dividing anything by 0 typically results in the function being undefined. Sal used the squeeze theorem with algebra to conclude that as x approaches 0 the limit oddly approaches 1. (Now that I’m thinking through this again, I’m realizing I don’t have a strong grasp on the function sin(x)/x and how/why it’s limit equals 1 as x approaches 0.)

After going through the Squeeze Theorem, I was then taught the names of three different types of discontinuities that can occur on a function. (I’m not sure if these are the only three types of discontinuities or if there are more that I haven’t learned about yet.) The term continuity in relation to a function just means that between a certain interval there is no ‘break’ in the function, i.e. the function is ‘continuous’, i.e. “within a continuous interval, you do not have to ‘pick up your pencil’ as you draw the function.” – Sal

The following are the three types of discontinuities I learned about:

1. Point/Removable Discontinuity 
   • A single point on a function where the value at that point is removed from the otherwise continuous flow of the function. At that point, f(x) could be a completely random value or it could be undefined.
     
2. Jump Discontinuity
   • This is when the function looks like it breaks apart at a specific section and ‘jumps’ up or down on the coordinate plane and continues on from there.
     
3. Asymptotic Discontinuity
   • This occurs when the function has an asymptote. I’m not sure if this only occurs when the asymptote is vertical or if it can also occur when an asymptote is on a diagonal, as well.

After working through discontinuities, I then learned (or maybe relearned) the notation for open and closed intervals:

• Open Interval
  • Uses rounded bracket to denote the interval. 
    • Ex. (-2, 1)
  • In an open interval, the interval is BETWEEN the numbers given and does not include the numbers as part of the interval.
    • i.e. x₁ < interval < x₂
      
• Closed Interval
  • Uses square brackets to denote the interval.
    • Ex. [-2, 1]
  • In a closed interval, the interval INCLUDES the numbers.
    • i.e. x₁ ≤ interval ≤ x₂

Lastly, I went through a few videos and exercises on how to calculate limits at infinity. I don’t think I’ve fully wrapped my head around this concept but, as far as I can tell, a limit at infinity can only be taken on a horizontal asymptote otherwise the limit at infinity is considered undefined. Here are two examples of limits that can be taken at infinity and the notation used for each example:

I learned that a limit can still be taken at infinity when dealing with trigonometric functions as long as the function oscillates across a horizontal line. I worked on questions that had me find the limit at infinity of functions that had square roots and different degreed terms in them. I didn’t find any of these questions too difficult to work through but I also don’t feel like I know them well enough to explain here. 

As a final side note, there were a few questions I went through this week that used logarithms and they started to become a bit easier for me to understand. In retrospect, I actually understood the gist of how they work but they still seemed unintuitive and hadn’t ‘settled‘ in my mind. Revisiting them this week, the general concept of how they worked seemed quite straightforward although I still have a somewhat hard time visualizing them in my head. Here are a few examples:

• Log₃(9) = 2
  • (This is the same thing as saying, “3 to the power of what equals 9? 2.”)
• Log₃(27) = 3
  • (“3 to the power of what equal 27? 3.”)
• Log₄(16) = 2
• Log₄(256) = 4
• Log₅(125) = 3
• Log(1,000,000) = 6
  • (When there’s no ‘base’ number (i.e. a number written in subscript between “Log” and the brackets) it’s implied that it’s base 10. This example would therefore be, “10 to the power of what equals 1,000,000? 6.”)
•  Log₇(2401) = 4

I’m glad that I only have 4 videos and 2 exercises left to get through in this unit, Limits and Continuity (2640/3500 M.P.) so I’m hoping I’ll have no problem finishing the unit early in the week. I feel pretty confident of my handle on everything that I’ve covered so far so hopefully it won’t take me too many attempts to get through the unit test. The following unit is called Derivatives: Definition and Basic Rules (0/2500 M.P.). I’ve heard the word ‘derivatives’ used when Sal or someone else talk about calculus and I’m quite sure that they’re a fundamental part of calc. I’m pretty pumped in a kind of nerdy way to finally learn what a derivative actually is! 🤓