The Benefits of Learning a Second Language

  • Context: MHB 
  • Thread starter Thread starter Lorena_Santoro
  • Start date Start date
  • Tags Tags
    Language
Click For Summary
SUMMARY

The discussion focuses on solving the limit of the expression 1/x - 1/sin(x) as x approaches 0 using L'Hôpital's Rule. The solution involves algebraic manipulation to express the limit as (sin(x) - x)/(x sin(x)). The derivatives of the numerator and denominator are calculated, leading to the conclusion that the limit is 0. This method, while not the simplest, demonstrates a creative approach to the problem.

PREREQUISITES
  • Understanding of limits in calculus
  • Familiarity with L'Hôpital's Rule
  • Basic knowledge of derivatives
  • Algebraic manipulation skills
NEXT STEPS
  • Study advanced applications of L'Hôpital's Rule
  • Learn about Taylor series expansions for limits
  • Explore alternative methods for solving limits
  • Practice problems involving derivatives and limits
USEFUL FOR

Students and educators in mathematics, particularly those focused on calculus and limit evaluation techniques.

Lorena_Santoro
Messages
22
Reaction score
0
 
Mathematics news on Phys.org
This may not be the simplest way to do it but the first thing I would do is do the algebra: 1/x- 1/sin(x)= sin(x)/xsin(x)- x/xsin(x)= (sin(x)- x)/xsin(x). Now use L'hopital. The derivative of sin(x)- x is cos(x)- 1 and the derivative of xsin(x) is sin(x)+ xcos(x). Both of those is 0 at x= 0 so do it again. Differentiating cos(x)-1 gives -sin(x). Differentiating sin(x)+ xcos(x) gives cos(x)+ cos(x)- xsin(x)= 2cos(x)-xsin(x). Finally, the numerator goes to 0 while the denominator goes to 2.

The limit is 0.
 
Country Boy said:
This may not be the simplest way to do it but the first thing I would do is do the algebra: 1/x- 1/sin(x)= sin(x)/xsin(x)- x/xsin(x)= (sin(x)- x)/xsin(x). Now use L'hopital. The derivative of sin(x)- x is cos(x)- 1 and the derivative of xsin(x) is sin(x)+ xcos(x). Both of those is 0 at x= 0 so do it again. Differentiating cos(x)-1 gives -sin(x). Differentiating sin(x)+ xcos(x) gives cos(x)+ cos(x)- xsin(x)= 2cos(x)-xsin(x). Finally, the numerator goes to 0 while the denominator goes to 2.

The limit is 0.

Indeed. This might not be the simplest way to go around it but it's really creative! Thanks and would be even greater if you comment below the video on Youtube so others can also get to know your way of solving it.
 

Similar threads

Book for self-learning calculus and learning in a foreign language?
  • · Replies 41 ·
2
Replies
41
Views
5K
Lingusitics What is the Easiest Spoken Language to Learn?
  • · Replies 25 ·
Replies
25
Views
3K
Learning data structures and algorithms in different programming languages
  • · Replies 86 ·
3
Replies
86
Views
2K
MHB We can find 3 scholars talking together (using one common language)
  • · Replies 3 ·
Replies
3
Views
2K
AI vs. foreign language
  • · Replies 6 ·
Replies
6
Views
2K
What's the most effective language for using AI?
  • · Replies 10 ·
Replies
10
Views
1K
Lingusitics Learning Target Language with Mother Tongue-subtitled Series
  • · Replies 4 ·
Replies
4
Views
1K
Insights FME in Probability - Conditionals in Natural Language - Comments
  • · Replies 2 ·
Replies
2
Views
2K
Call program written in language B from program written in language A
  • · Replies 22 ·
Replies
22
Views
3K
Undergrad What is the Definition of a Sequence in Formal Language Theory?
  • · Replies 4 ·
Replies
4
Views
2K