Provide an appropriate response.
-If $\lim f ( x ) = 1 , \lim _ { x \rightarrow 7 ^ { + } } f ( x ) = - 1$, 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) II and III only
B) I, II, and III
C) I and II only
D) I and III only

Question

Quizplus · Accepted Answer

Since $f(x)$ is an even function, we know that $\lim_{x\rightarrow 1^{-}}f(x)=\lim_{x\rightarrow -1^{+}}f(x)=\lim_{x\rightarrow -1^{-}}f(x)=\lim_{x\rightarrow 1^{+}}f(x)$. Also, we know that $\lim_{x\rightarrow 7^{+}}f(x)=-1$, so $\lim_{x\rightarrow -7^{-}}f(x)=-1$. Since $\lim_{x\rightarrow-\infty}f(x)=\lim_{x\rightarrow\infty}f(x)=1$, the graph of $f(x)$ has to look like this:

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    /\
   /  \
  /    \
 /      \       
/--------\

Since $\lim_{x\rightarrow 1^{-}}f(x)=-1$ and $\lim_{x\rightarrow -1^{+}}f(x)=1$, the graph must have a vertical asymptote at $x=1$ and $x=-1$, respectively. Since the graph is symmetric about the y-axis (because $f(x)$ is an even function), there must also be a vertical asymptote at $x=-1$. Therefore, $\lim_{x\rightarrow 1}f(x)$ does not exist (because the left and right limits are not equal), and the only statements that are true are I and III.

Provide an Appropriate Response lim⁡f(x)=1,lim⁡x→7+f(x)=−1\lim f ( x ) = 1 , \lim _ { x \rightarrow 7 ^ { + } } f ( x ) = - 1limf(x)=1,limx→7+​f(x)=−1

Provide an Appropriate Response $\lim f ( x ) = 1 , \lim _ { x \rightarrow 7 ^ { + } } f ( x ) = - 1$