Question

lim cosx - sinx/ pi/4-x , x-->pi/4

Answer 1

Lim(x→π/4) { cosx - sinx}/(π/4 -x)

put x = π/4
you see limit in the form of 0/0
so , we use L-Hospital rule
here

differentiate numerater and denominator wrt x

then ,
Lim(x→π/4) { -sinx - cosx }/-1
now put x = π/4

then limit value = √2

2nd method -
===°==========
cosx -sinx = -√2sin(x -π/4)
hence,
Lim(h→0) { -√2sin(x -π/4)/(π/4 -x)}

x→π/4

let x = h+π/4

then
h→0

and limit convert as
Lim(h→0) { -√2sin(h+π/4-π/4)}/(π/4 -π/4 -h)

Lim(h→0) √2sinh/h

we know,
Lim(x→0) sinx/x = 1

so, √2 Lim(h→0)sinh/h = √2 × 1