Evaluate the following limit: `lim_(x->pi/4)(sqrt(2)-cos x-sin x)/((pi/4-x)^2)`

Explanation

`lim_(x->pi/4)(sqrt(2)-cos x-sin x)/((pi/4-x)^2)`

This is `(0/0)` form

So apply L-Hospital Rule

`lim_(x->pi/4)(d/(dx)(sqrt(2)-cos x-sin x))/(d/(dx)((pi/4-x)^2))`

`lim_(x->pi/4) (sin x-cosx)/(2(pi/4-x))`

`lim_(x->pi/4) (cos x+ sinx)/(-2)`

`=(1/(sqrt2)+1/(sqrt2))/(-2)`

`=-1/(sqrt2)`