Question

Question #004d9

Answer 1

`lim_(xrarr2)(cos(pi/x))/(x-2)=pi/4`

Explanation:

`lim_(xrarr2)(cos(pi/x))/(x-2)`

Attempting to evaluate the limit gives `cos(pi/2)/(2-2)=0/0`, which is an indeterminate form. Thus, we can use l'Hopital's rule, which says to take the derivative of the numerator and denominator separately:

`=lim_(xrarr2)(d/dx(cos(pi/x)))/(d/dx(x-2))`

`=lim_(xrarr2)(-sin(pi/x)d/dx(pix^-1))/1`

`=lim_(xrarr2)(-sin(pi/x))(-pix^-2)`

`=lim_(xrarr2)(pisin(pi/x))/x^2`

The limit can now be evaluated:

`=(pisin(pi/2))/2^2`

`=pi/4`