Question

Plz answer this question!!!! ​

Answer 1

Step-by-step explanation:

Add and subtract sin(π/4) = cos(π/4) = 1/√2 in the numerator:

[cos(x) - sin(x)]/(x - π/4)

= [cos(x) - 1/√2 + 1/√2 - sin(x)]/(x - π/4)

= [cos(x) - cos(π/4) + sin(π/4) - sin(x)]/(x - π/4)

= [{cos(x) - cos(π/4)} - {-sin(π/4) + sin(x)}]/(x - π/4)

= [{cos(x) - cos(π/4)} - {sin(x) - sin(π/4)}]/(x - π/4)

= [cos(x) - cos(π/4)]/(x - π/4) - [sin(x) - sin(π/4)]/(x - π/4)

If x is approaching π/4, this is the derivative of cos(x) at π/4 minus the derivative of sin(x) at π/4, so we have:

lim(x→π/4) [cos(x) - sin(x)]/(x - π/4)

= lim(x→π/4) {[cos(x) - cos(π/4)]/(x - π/4) - [sin(x) - sin(π/4)]/(x - π/4)}

= lim(x→π/4) [cos(x) - cos(π/4)]/(x - π/4) - lim(x→π/4) [sin(x) - sin(π/4)]/(x - π/4)

= d/dx[cos(x)]_(at x=π/4) - d/dx[sin(x)]_(at x=π/4)

= [-sin(x)]_(at x=π/4) - [cos(x)]_(at x=π/4)

= -sin(π/4) - cos(π/4)

= -1/√2 - 1/√2

= -√2

hope it will work  !