Use the continuity to evaluate the given limit

Question

Use continuity to evaluate the limit. lim x→π cos(x + cos x)

Answer 1

Method 1:

lim x→π cos(x +cos (x)) = cos(π +cos(π)) = cos(π-1) because cos is everywhere continuous.

Method 2:

According to the cosines addition formula we have: cos(x + cos(x))= cos(x).cos(cos(x)) - sin(x) sin(cos(x)). Since cos and sin are everywhere continuous, we could simply plug in the limit π in the equation to obtain:

lim x→π cos(x +cos (x)) = cos(π).cos(cos(π)) - sin(π) sin(cos(π)) = cos(π) cos(-1) - 0 = cos(π) cos(-1) = cos(π-1)