MHB The Benefits of Learning a Second Language

[Lorena_Santoro]

 

[HOI]

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.

 

[Lorena_Santoro]

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.