Calculus and Its Applications, Chapter 1, 1.2, Section 1.2, Problem 38

The function

$C(x) = \left\{ \begin{equation} \begin{aligned} &-1, && \text{for } x < 2,\\ &1, && \text{for } x \geq 2 \end{aligned} \end{equation} \right.$





(a) Find $\displaystyle \lim_{x \to 2^+} C(x)$
Based from the graph, $\displaystyle \lim_{x \to 2^+} C(x) = 1$

(b) Find $\displaystyle \lim_{x \to 2^-} C(x)$
Based from the graph, $\displaystyle \lim_{x \to 2^-} C(x) = -1$

(c) $\displaystyle \lim_{x \to 2} C(x)$
Based from the graph, $\displaystyle \lim_{x \to 2} C(x)$ does not exist the
$\displaystyle \lim_{x \to 2^-} C(x)$ is not equal to the $\displaystyle \lim_{x \to 2^+} C(x)$

(d) Find $C(2)$
Based from the graph, $C(2) = 1$

(e) Is $C$ continuous at $x =2$? Why or why not?
No, because the $\displaystyle \lim_{x \to 2} C(x)$ does not exist

(f) Is $C$ continuous at $x = 1.95$? Why or why not?
Yes, because the $\displaystyle \lim_{x \to 2^-} C(x)$ exist