If `f(x)=(sqrt(2)cosx-1)/(cotx-1)` , `x!=pi/4` . Find the value of `f(pi/4)` so that `f(x)` becomes continuous at `x=pi//4` .

Explanation

f(x) continuous at `x = pi/4` if

`lim _{x rightarrow pi / 4} f(x)=f(frac{pi}{4})`


`Rightrightarrow f(frac{pi}{4})=lim _{x rightarrow pi / 4} frac{sqrt{2} cos x-1}{cot x-1}`


`=lim _{x rightarrow pi / 4} frac{(sqrt{2} cos x-1) sin x}{cos x-sin x}`


`=lim _{x rightarrow pi / 4} frac{(sqrt{2} cos x-1)(sqrt{2} cos x+1)}{(sqrt{2} cos x+1)(cos x-sin x)} cdot frac{(cos x+sin x)}{(cos x+sin x)} cdot sin x`


`=lim _{x rightarrow pi / 4}(frac{2 cos ^{2} x-1}{cos ^{2} x-sin ^{2} x}) frac{(cos x+sin x) sin x}{(sqrt{2} cos x+1)}`


`=lim _{x rightarrow pi / 4} frac{sin x(cos x+sin x)}{sqrt{2} cos x+1}`


`=frac{frac{1}{sqrt{2}}(frac{1}{sqrt{2}}+frac{1}{sqrt{2}})}{sqrt{2} cdot frac{1}{sqrt{2}}+1}=frac{1}{2}`