When you 'substitute' , you get:

undefined.

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

= undefined.

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

= 0.

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.

ILoveMaths07.