Provide an appropriate response.
-Given $\lim _ { x \rightarrow \theta ^ { - } } f ( x ) = L _ { 1 } , \lim _ { x \rightarrow \theta ^ { + } } f ( x ) = L _ { r }$, and $L _ { 1 } \neq L _ { r }$, which of the following statements is true?
I. $\lim _ { x \rightarrow - } f ( x ) = L _ { 1 }$
II. $\lim _ { x \rightarrow \theta } f ( x ) = L _ { r }$
III. $\lim _ { x \rightarrow 0 } f ( x )$ does not exist.
A) none
B) III
C) II
D) I

Question

Quizplus · Accepted Answer

The correct statement is III because for a limit to exist at a point, the left-hand limit and the right-hand limit must be equal. Since $L_1 \neq L_r$, the limit of $f(x)$ as $x$ approaches $\theta$ does not exist, making all specific statements about the limit at $\theta$ incorrect. The question does not provide information about the limit as $x$ approaches 0, so III is a misinterpretation of the given information. However, the intended correct answer seems to be based on the premise that the limit does not exist at $\theta$, not 0. The correct interpretation should focus on the limit at $\theta$, not 0, indicating a misunderstanding in the explanation.

Provide an Appropriate Response lim⁡x→θ−f(x)=L1,lim⁡x→θ+f(x)=Lr\lim _ { x \rightarrow \theta ^ { - } } f ( x ) = L _ { 1 } , \lim _ { x \rightarrow \theta ^ { + } } f ( x ) = L _ { r }limx→θ−​f(x)=L1​,limx→θ+​f(x)=Lr​

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