Question

calculate: lim as x approaches pi/4 = [(x - pi/4)^2 / (tan x - 1)^2] .

Answer 1

the first thing that should pop up in the mind is to substitute
\(u = x-\pi/4 \\ as, x\to \pi/4, u\to 0 \)

make this substitution!
and tell me what u get ?

Answer 2

lim as u approaches 0 = [(u^2) / (tan x-1)^2] ?

Answer 3

you have to change tan x as well! :)

as, u = x- pi/4
x = u + pi/4
right ?

so tan x changes to

tan (u+ pi/4)
got it ?

Answer 4

ya :D. then?

Answer 5

expand tan (u+pi/4)

do you know the formula for
tan (A+B)
?

Answer 6

also put tan pi/4 = 1 :)

Answer 7

it will be : [(tan x +tan pi/4)/ (1-tan x.tan pi/4)] ?

Answer 8

sorry, imean it will be : [(tan u +tan pi/4)/ (1-tan u.tan pi/4)] ?

Answer 9

yes

Answer 10

now put tan pi/4 = 1

Answer 11

(tan u + 1)/(1-tan u) ?

Answer 12

yes
now we had tan x -1 in the denominator
so can you try to simplify

(tan u + 1)/(1-tan u) -1 = ... ?

Answer 13

will it be (2 tan u)/(1-tan u) ?

Answer 14

correct!
plug this in your function

Answer 15

(u^2)/(2tan u/1-tan u)^2 ..