Laboratory Project: An Elusive Limit

2개월 전


In this video I go over the Laboratory Project titled An Elusive Limit. Laboratory Projects are interesting math projects at the end of some sections of my calculus book, and in this case at the end of the section on Taylor and Maclaurin Series. The project I cover involves solving the limit of a function that has a difference in the numerator and denominator; and both include trigonometric sine and tangent functions, as we well as inverse trigonometric sine and tangent functions.

The interesting part of all of these functions is that as x approaches 0, then they all approach 0 in about the same rate so that their values are near identical to a high level of digits precision. This makes the resulting differences in the numerator and denominator of the function cancel out most of the digits and thus we are left with only a few correct decimal places. This makes the resulting calculations highly dependent on the level of precision of the calculator being used, and in most cases the calculator won’t be able to brute force calculate the function at values very close to x = 0.

In this case we can still solve the limit in several ways, either by using a computer algebra system (CAS) to solve derivatives or power series of the numerator and denominator, and thus be able to use methods such as l’Hospital’s Rule or other limit laws and algebra manipulation. This effectively means that if we can’t compute the function directly we can instead focus on simplifying or transforming the numerator and denominator into their equivalent but different formulations, which may allow for the limit to be solved through algebraic or calculus means.

This is a very interesting tutorial involving the use of calculators and understanding the levels of precision to be certain that the calculations are correct, so make sure to watch this video!

The topics covered in this video are listed below with their time stamps.

  • @ 1:12 - Topics to Cover
  1. @ 1:33 - Laboratory Project: An Elusive Limit
    • 6 Questions
  2. Solutions
    • @ 3:13 - Solution to Question 1
    • @ 19:51 - Solution to Question 2
    • @ 21:14 - Solution to Question 3
    • @ 28:40 - Solution to Question 4
    • @ 41:12 - Solution to Question 5
    • @ 43:13 - Solution to Question 6

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Laboratory Projects Playlist: https://www.youtube.com/playlist?list=PLai3U8-WIK0GPz_diCZTkiVs6Q4mA_KHb

Laboratory Project: Logistic Sequences: https://peakd.com/mathematics/@mes/laboratory-project-logistic-sequences

Sequences and Series Playlist: https://www.youtube.com/playlist?list=PLai3U8-WIK0EXHAJ3vRg0T_kKEyPah1Lz

Infinite Sequences and Series: Taylor and Maclaurin Series: https://peakd.com/mathematics/@mes/infinite-sequences-and-series-taylor-and-maclaurin-series .

Limits Playlist: https://www.youtube.com/playlist?list=PLFE39AF0D7E0651AC


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In this video I go through an in-depth analysis of a function whose limit pushes beyond the digits of precision of most calculators.

View video notes on the Hive blockchain: https://peakd.com/hive-128780/@mes/laboratory-project-an-elusive-limit