MHB The Benefits of Learning a Second Language

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The discussion focuses on a mathematical approach to finding the limit of a specific expression involving sin(x) and x. The method involves using algebraic manipulation followed by L'Hôpital's Rule to differentiate the numerator and denominator. After several iterations of differentiation, the final result indicates that the limit approaches 0. Participants appreciate the creativity of the solution, suggesting it could benefit others if shared on platforms like YouTube. Overall, the conversation highlights an unconventional yet effective technique for solving the limit problem.
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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.
 

Thread 'Area of Overlapping Squares'

Here is a little puzzle from the book 100 Geometric Games by Pierre Berloquin. The side of a small square is one meter long and the side of a larger square one and a half meters long. One vertex of the large square is at the center of the small square. The side of the large square cuts two sides of the small square into one- third parts and two-thirds parts. What is the area where the squares overlap?
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