An Elusive Limit Question 3: Using L'Hospital's Rule 7 Times in a Row (Using Wolfram Alpha)

mes (67)in DTube • 5 days ago

In this video I go over the previous difficult trigonometric (and inverse trigonometric) function and this some apply L'Hospital's Rule to show that its limit as x approaches 0 is equal to 1. When plugging in x = 0 into the numerator and denominator of the function, we get the indeterminate form 0/0. This means we can apply L'Hospital's Rule and take the limit of the derivatives of the numerator and denominator. Since the question asked us to use a Computer Algebra System (CAS), I have calculated the derivatives using the amazing Wolfram Alpha online calculator: https://www.wolframalpha.com/

The first 6 derivatives or applications of L'Hospital's Rule all obtain the indeterminate form 0/0. But the 7th derivatives obtain the limits -168/(-168) = 1.

Time stamps:

Question 3: Evaluate limit with l'Hospital's Rule: 0:00
Solution: Limit is in an indeterminate form: 0:22
Recap l'Hospital's rule: 2:16
Using Wolfram Alpha, the first 6 derivatives are indeterminate but 7th is non-zero: 3:25
The limit is equal to 1: 5:46

Full video below: