Question

show that f(x)=cos x is continuous everywhere

Answer 1

\[\lim_{h \rightarrow 0}\cos(x+h)=\lim_{h \rightarrow 0}\cos(x-h)=\cos(x) \forall x \in R\]

Answer 2

In other words, for any x value, as we approach x from either side, cos x approaches the same value (same limiting value), which implies continuity. Only if cos x were to approach a different value as we came from the left or the right towards x would cos x be discontinuous.

Answer 3

A root of an equation is an x value that makes the equation true.