Question

Evaluate: `lim_(xto(5pi)/(4)) [sinx+cosx]`, [.] denotes the greatest integer function.

Answer 1

We have,
`L=underset(xto(5pi)/(4))lim[sinx+cosx]=underset(xto(5pi)/(4))lim[sqrt(2)sin(x+pi/4)]`
Now, `underset(xto(5pi)/(4)).[sqrt(2)sin(x+(pi)/(4))]=[sqrt(2)sin((3pi^(+))/(2))]=[-sqrt(2)]=-2`
And `underset(xto(5pi^(+))/(4))lim[sqrt(2)sin(x+(pi)/(4))]=[sqrt(2)sin((3pi^(-))/2)=[-sqrt(2)]=-2`
So, `L=-2`