Evaluate the limit (DO not use L'Hospital Rule)

Question

lim x---> infinity arctan(cos(1/x))

Answer 1

1/x approaches zero as x approaches infinity

cos(1/x) =cos(0)=1 arctan(1)=45 degrees = pi/4

the angle whose tangent = 1 is 45 degrees or pi/4

the tangent is opposite side over adjacent side, in a right triangle. When tangent = 1 that means

opposite side=adjacent side. That's a right triangle with two 45 degree angles. a right triangle has a 90 degree angle, and if the other two sides are equal then the other two angles are equal. All 3 angles sum to 180, so 180-90= 90 = sum of the other two angles. Since they're equal 90/2 = 45 degrees for each of those two angles.