An Elusive Limit Question 4-5: Evaluating the Limit using Taylor Series (and Wolfram Alpha)

mes (67)in DTube • 3 days ago

In this video I go over the earlier trigonometric (and inverse trigonometric) limit and solve it by writing the numerator and denominator as Taylor series. Using the incredibly powerful Wolfram Alpha online calculator to first determine the Taylor series, I show that they cancel out to obtain the limit equal to 1. Both Taylor series start off with the same x^7 term, which means we can also apply l'Hospital's Rule 7 times to obtain the same answer, just like in Question 3. Finally I solve the entire limit directly using Wolfram Alpha, which magically solves it in a fraction of a second!

Time stamps:

Question 4: Evaluating numerator and denominator using Taylor series: 0:00
- Solution: Solve Taylor Series using Wolfram Alpha: 0:36
- Big O Notation indicates the remaining terms of a series is at max a constant multiplied by the term inside: 2:00
- Evaluating the limit using the Taylor series: 3:04
- Limit is equal to 1: 7:04
- Can apply l'Hospital's Rule 7 times to obtain the same result as Question 3: 7:38
- Calculation check matches Question 3: 12:08
Question 5: Solve for the limit directly using Wolfram Alpha: 12:22

Full video below: