Question

. For this problem, you may find Proposition 3.1.5 useful which in turn implies that tan x is continuous whenever x is not an odd multiple of π 2 . Moreover, you can assume that sin x and cos x are positive and negative on appropriate intervals. Let I := (− π 2 , π 2 ). (a) Show that tan x is strictly increasing and, hence, injective on I. (b) Prove that lim x→− π 2 + tan x = −∞ and lim x→π 2 − tan x = ∞ Use this to conclude that f(I) = R. You may find Exercise 6 in Section 3.7 useful here. (c) Show that arctan x = tan−1 x maps R to I and is differentiable everywhere with d dx tan−1 x = 1 1 + x 2 for all x ∈ R. (d) Prove that limx→∞ tan−1 x = π 2 and lim x→−∞ tan−1 x = − π 2

Answer 1

Thank you.