When you 'substitute' , you get:
So, we have to use L'Hopital's Rule. Differentiating, you get:
Uggh, for some reason, there's a Syntax Error in my Latex Code, and I'm in a hurry, so I'll just type it out normally.
Differentiating, you get:
lim x ---> infinity (x . cos (pi/x) . (-pi/x^2) + sin (pi/x)) ---> (Product Rule)
'Plugging in' infinity now gives:
(infinity. cos (pi/infinity) . (-pi/infinity^2) + sin (pi/infinity))
= (infinity. cos 0 . 0 + sin 0)
= (infinity. 1. 0 + 0)
= infinity. 0 + 0
So, keep differentiating and 'plugging in' infinity into each derivative until you get a definite answer.
For the second one,
Plug in x = pi/4 first.
You get (1 - tan (pi/4)) * sec (pi/4)
= (1 - 1) * (1 / cos (pi/4))
= 0 * (1 / ((sqrt (2) / 2)))
= 0 * sqrt 2
So the limit, as x approaches pi/4, of (1 - tan x) * sec x = 0.
I hope that helps.
Please forgive me if I've made any errors.