Single Variable Calculus, Chapter 2, 2.3, Section 2.3, Problem 50

Given the function $f(x) = [[\cos x]], - \pi \leq x \leq \pi$

(a) Graph the function $f(x)$















(b) Find each limit, if it exists.


$
\begin{equation}
\begin{aligned}
(i) & \lim \limits_{x \to 0} f(x) &
(ii) & \lim \limits_{x \to (\pi/2)^-} f(x)\\

(iii) & \lim \limits_{x \to (\pi/2)^+} f(x) &
(iv) & \lim \limits_{x \to(\pi/2)} f(x)
\end{aligned}
\end{equation}
$


$(i)$ Referring to graph given, the $\lim \limits_{x \to 0} f(x) = 0 $

$(ii)$ Referring to graph, the $\displaystyle \lim \limits_{x \to (\pi/2)^-} f(x) = 0$

$(iii)$ Referring to graph, the $\displaystyle \lim \limits_{x \to (\pi/2)^+} f(x) = -1$

$(iv)$ The $\displaystyle \lim \limits_{x \to (\pi/2)} f(x)$ does not exist because the left and right hand limit are different.

(c) Find the values of $a$ that $\lim \limits_{x \to a} f(x)$ exist.

Referring to the graph, we can say that there are no values of $a$ that would make the limit of $f(x)$ exist
because of the difference between its left and right hand limits.