Provide an appropriate response.
-If $\lim _ { x \rightarrow 1 ^ { - } } f ( x ) = 1$, $\lim _ { x \rightarrow 1 ^ { + } } f ( x ) = - 1$, and $f ( x )$ is an even function, which of the following statements are true?
I. $\lim _ { x \rightarrow 1 ^ { - } } f ( x ) = - 1$
II. $\lim _ { x \rightarrow 1 ^ { + } } f ( x ) = - 1$
III. $\lim _ { x \rightarrow 1 } f ( x )$ does not exist.
A) I and III only
B) I and II only
C) I, II, and III
D) II and III only

Question

Quizplus · Accepted Answer

Since $f(x)$ is an even function, we know that $f(-1)=f(1)=0$. We also know that $\lim_{x\rightarrow 1^-}f(x)=1$. Since $f(x)$ is even, $\lim_{x\rightarrow -1^+}f(x)=\lim_{x\rightarrow 1^-}f(x)=1$. 
But $\lim_{x\rightarrow 1^+}f(x)=-1$, so by definition, $\lim_{x\rightarrow 1}f(x)$ does not exist. Thus, only statements I and III are true.

Provide an Appropriate Response lim⁡x→1−f(x)=1\lim _ { x \rightarrow 1 ^ { - } } f ( x ) = 1limx→1−​f(x)=1

Provide an Appropriate Response $\lim _ { x \rightarrow 1 ^ { - } } f ( x ) = 1$