It was a bit of a weird week. I had to spend ~50% of the week reviewing trigonometric identities which, by the end of it, I started to get a better picture of but still haven’t figured them out completely. I also had to combine using trig identities with algebra to solve for limits which I found pretty difficult to understand. For the most part, however, the majority of what I went through this week was relatively easy to figure out and as enjoyable as learning math can be. I’ve learned how helpful it can be to Google other non-KA related videos when I get stuck on something to try and get a broader picture of what I’m struggling with. I did this to understand the Sine and Cosine Sum identity this week and, after not understanding it for months, it finally started to click for me. Googling other resources/teachers when I get stuck on something will be something I do more regularly, moving forward.

Everything I covered this week revolved around determining limits using the following flow chart from KA:

In the flow chart, you see that there are 7 steps from (A) to (G). The first step to solve a limit is simply inputting the value of x into the function f(x). If the output results in b/0, as indicated in step (B), this is typically because the x-value falls on vertical asymptote. In my mind, this makes sense since slope is calculated as “rise/run” so if the run equals 0 then the line created would have nowhere to go but straight up. (Not sure if that made sense…) Here’s a photo from Desmos where I graphed the function f(x) = x/(x – 1):

If you input a value into f(x) and the output is a real number, as indicated in step (C), that output is typically the limit. (I’m actually not completely sure what the exceptions are to this rule of thumb.)

Finding the limit gets tricky when the output equals 0/0, as indicated in step (D). When the output equals 0/0, this is known as indeterminate form (read: the limit cannot be determined). If the output is in this form, you must then move on to steps (E), (D), or (G):

• Step (E) – Factoring
  • Uses algebraic manipulation to factor the polynomials in the numerator and/or the denominator and then cancel out equal factors which can change the quotient from 0/0 to something else where you can find the limit. If that’s the case, when writing the answer you must remember to state that “x ≠ (whatever value was leading to 0/0)”. (I just realized I didn’t do that in the photo below. 🤦🏻‍♂️) 

• Step (F) – Using Conjugates
  • This method is used when there’s a radical in either the numerator or the denominator of the function and the radical is part of a binomial. With this technique, you multiply both the numerator and denominator by the exact same binomial that contains the radical EXCEPT the (+/-) sign in front of the second term is switched (a.k.a. the binomial’s conjugate). Even though you’re still left with a radical (most of the time? All the time? I’m not sure.), you can often find the limit with the new quotient.

• Step (G) – Trig Identities
  • This was by far the hardest of the three methods for me to work through, mainly because I don’t have a solid grasp on trig identities. This method only works if the function contains trigonometry. If it does and the quotient of the function equals 0/0, you can sometimes switch the part of the function that contains the sine, cosine, etc. value with another trig identity which can then let you solve the new equation. (I don’t think that made sense…)

To better understand how to solve for limits that contained trigonometry and required using trig identities, I had to go back and watch a video where Sal went through how to derive the different identities. Although I didn’t find it too difficult to follow along with his algebra, I definitely don’t understand the identity derivations well enough to explain them myself. Regardless, here’s a list of the identities Sal derived from that video:

After I watched the video going through the above identities, I was annoyed that I didn’t understand the fundamental Sine and Cosine angle sum identity. I Googled it and came across this video which really helped me to understand it:

Again, I’m not going to explain the step-by-step process of what the guy in the video does but here’s a photo from my notes of the same concept:

I’m still not 100% clear how the Sine and Cosine sum identity works based off this example but the video definitely gave me a better grasp of the concept, in general. One thing I don’t understand is what happens if the angles add up to more than 90 degrees. Nonetheless, bringing this all back to solving limits with trig identities, it seems like as long as I can recognize which trig identity to use to at which time I’ll be able to solve the question even though I don’t fully understand what’s going on. (This is definitely one of my biggest pet peeves though, when I just use a formula without fully understanding it. 😡)

I’m now 46% of the way through Limits and Continuity (1600/3500 M.P) so I would say it’s definitely still possible for me to get through this unit by the end of the month. Units tend to get harder as I get further into them, however, so I won’t be surprised if my pace continues to slow down as I go through it. Of the remaining 12 exercises, it looks like only one of them uses trig so hopefully that means I won’t struggle as much to get through the rest of them. 🤞🏼