Since the last course I worked through, Precalculus, was mostly all review, it’s been a long time since I worked through a course that taught me new concepts (mid-December to be exact). Getting started with Calculus 1 this week, it was nice getting back into a groove of slowly working through new concepts and trying to wrap my head around them. I enjoy figuring out which concepts are worth spending a bit of extra time on to write out meticulous notes and diagrams to make them look good. I also enjoyed the exercises I worked through this week as they weren’t too difficult or frustrating yet I still learned a number of semi-tricky concepts. I also ended up getting ~33% of the way through the unit which means it may only take me ~3 weeks to get through the unit as opposed to the ~2 months I thought it would. WOOO! 🥳 🎉🎊

I began this week learning that there’s a way to denote a limit when it approaches from the left of a point (i.e. from the negative side of the x-axis) and from the right of a point (i.e. the positive side of the x-axis):

In the function shown in the top left corner of the page, you can see that at x = 2, f(x) = 3. You can also see that as the function approaches x= 2 from the left, f(x) approaches 4 and as the function approaches x = 2 from the right, f(x) approaches 2. From what I’ve learned so far, this means that the limit of f(x) does not exist at x = 2 BUT that you can say that a limit exists at x = 2 when approaching from just the left or just right and it would equal 4 and 2 respectively. The notation used to denote this concept is written out in the bottom half of the page. The difference is there’s either a + or – sign written in superscript beside the number the x-value in question that’s written below “lim”.

Most of the time if a limit doesn’t exist Sal has either literally used the phase “the limit does not exist” or has said that the limit is “undefined”, but there seems to be another word that he uses, which is “unbounded”, specifically if the limit of a point in question lands on a asymptote:

I’m not exactly sure why the word unbounded is used with asymptotes or what the exact definition of unbounded is at this point. I don’t have a very intuitive understanding of how asymptotes work either so I’m hoping as I go through calc I’ll get a better idea of both.

Next, I learned about what Sal described as the “formal definition” of a limit. In layman’s terms, my understanding is that it states there’s a range on the x-axis that overlaps the point in question which corresponds to a range on the y-axis which overlaps the limit. (Reading that back, I’m sure I’m missing something and not explaining it very well but that’s my understanding of the formal definition of a limit so far.)

In the above photo, there are four variables that need to be defined:

• L – “Limit”
  • The point in question on the y-axis where the limit approaches from both the negative and positive directions but never actually reaches.
    
• C – “Point C”
  • The point on the x-axis which corresponds with the limit in question on the function.
    
• ∈ – “Epsilon”
  • Denotes the range of the limit on the y-axis. The entire range, from above and below L, would equal 2∈.
    
• δ – “Delta”
  • Denotes the range on either side of Point C on the x-axis. The entire range, from the left of C and the right of C, would equal 2δ.

The formula for this definition is about 2/3 of the way down the page which is |x – C| < δ —> |f(x) – L | < ∈. I don’t really understand this formula right now or understand its purpose but, again, I’m hoping I’ll figure both of those things out as I get further into calc.

I was then introduced to what are known as “Limit Properties” which didn’t seem too difficult to wrap my head around. These are essentially formulas you use to add, subtract, multiply, divide limits to and from each other and also to raise limits to powers. I was glad that each property seemed fairly straightforward. The most difficult parts of these properties was wrapping my head around the idea of combining limits (which still doesn’t make intuitive sense to me) and the notation used for each property:

The last thing I learned this week was that it’s possible to combine two limits on two separate functions that are undefined individually. Here’s a page from my notes that explains this concept and a screen shot of a question I did:

My general understanding of this concept is that if the limits on each individual function are undefined but the limits on the left side of both functions equal the same as the limits on the right side of both functions when they’re added/subtract to or from each other, then the limit exists. (Once again, that seems like a pretty bad explanation of this concept works but I really don’t understand it well enough to explain it much better than that at this point.)

As I mentioned at the beginning of this post, I’m already ~1/3 of the way through this unit, Limits and Continuity (1,120/3,500 M.P.), so I’m really hoping I can finish it by the end of the month. There’s always a possibility that I’m introduced to something that I find super confusing that takes me a long time to understand so I won’t be surprised if it takes me longer than a month to get through this unit. Regardless, that’s my new goal. If I could somehow manage the pace of getting through 1 unit a month, I would be able to get through the entire course by the end of November. That seems unlikely but maybe getting through the course by the end of the year could be doable. That would be dope af.

Also, I just passed 100,000 words from my blog posts, let’s goooo!