Learning about the definition of limit and the limit theorems felt fresh to the mind. The concept of limits had already been introduced before, but learning about several theorems made understanding it a tad bit easier.
What’s important about learning the theorems regarding limits is that we can be guided in solving limit problems.
In my opinion, I would dare to say that I did good in learning about these concepts, but it does not mean that I am confident that I could solve all problems that would cross my path. This just means that I should further enhance my knowledge by having the right amount of practice.
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DEFINITION OF LIMITS
LIMIT THEOREMS
Learning guides 3.1 and 3.2
Hello! Welcome back to this blog! For this week, we have already started with the lessons in second quarter. This means that we will face much complex topics than before. I am here to help you understand some problems regarding the topics Definition of Limits and Limit theorems. So, let’s begin!
Learning guide 3.1.2 (Definition of Limits)
In this learning guide, both the intuitive and formal definitions of limits were introduced and as an exercise, we will be proving the limits below if they are true based on the formal definition of limits.
We were able to prove the given limits based on the formal definition of limits.
Learning guide 3.2.1 (Limit theorems)
There are theorems that we can use in evaluating limits of functions algebraically. In this learning guide, nine theorems were introduced and we will be using these theorems to evaluate the limit of the function given.
In evaluating the limit of the function above, we used long version where we used theorems 2 (the limit of a constant is itself), 3 (the limit of x as x approaches a is a), 4 (the product of a constant and a function is equal to the product of the limit of the function and the constant), 5 (the limit of the sum of functions is equal to the sum of the limits of each function), and 8 (the limit of an integer power of a function is equal to the power of the limit of the function). The other method of solving this would be inputting the a value to the function = f(a).
Learning guide 3.2.2 (Limit theorems)
In learning guide 3.2.1, the nine limit theorems were introduced. However we cannot directly use these theorems for some cases such as the indeterminate type of 0/0. In this learning guide, we will be solving examples of this indeterminate type of 0/0.
As we can see, when 7 was substituted as x, the limit would be 0/0. For this case, we can factor out the numerator to cancel the denominator which leads us to x^2-1. From there, we can now use the limit theorems in Learning guide 3.2.1 to evaluate the limit.
In this example, we have a radical in the numerator that is why we need to rationalize the numerator by multiplying the conjugate. After multiplying we can see that the term 2x-13 (present in both the numerator and denominator) can be cancelled, which will equal to 1. From there, we can use the limit theorems to evaluate the limit.
Reflection (Week 1)
I have learned a lot from my habits during the first quarter and, I have identified the areas in which I can improve more. As we start a new quarter, I vow to improve as a student. I will apply what I think would be best for me in studying. When I browsed the learning guides for this week, at first, I was overwhelmed because I thought the topics for this quarter are much complicated than before. However, as I read and study the material, I gained confidence in solving items related to the topic. Although I don’t know yet what I got in the quiz, I am confident that I have fully understood the lesson. I will maintain my good habits during the first week.
I know that I am off to a good start this quarter.
