Question

Question #6c4e4

Answer 1

`Lim_(xrarr1)(sinx-sin(1/x))/(e^x-e^(1/x))~~0,1987`

Explanation:

`Lim_(xrarr1)(sinx-sin(1/x))/(e^x-e^(1/x))=0/0`

Using L'Hopitals rule:

`Lim_(xrarr1)(sinx-sin(x^-1))/(e^x-e^(x^-1))`

`Lim_(xrarr1)(cosx-cos(x^-1)*-1*x^-2)/(e^x-e^(x^-1)*-1*x^-2)`

`Lim_(xrarr1)(cosx+(cos(x^-1))/x^2)/(e^x+(e^(x^-1))/x^2)`

`(cos1+cos1/1^2)/(e+e/1^2)=(cancel2cos1)/(cancel2e)=cos1/e~~0,1987`