Provide an appropriate response.
-Given $\lim _ { x \rightarrow \theta ^ { - } } f ( x ) = L , \lim _ { x - \theta ^ { + } } f ( x ) = L _ { r }$, and $L _ { 1 } = L _ { r }$, which of the following statements is false?
I. $\lim _ { x \rightarrow 9 } f ( x ) = L _ { l }$
II. $\lim _ { x \rightarrow \theta } f ( x ) = L _ { r }$
III. $\lim _ { x \rightarrow \theta } f ( x )$ does not exist.
A) III
B) II
C) none
D) I

Question

Quizplus · Accepted Answer

For the limit of $f(x)$ as $x$ approaches $\theta$ to exist, the left-hand limit and the right-hand limit as $x$ approaches $\theta$ must be equal. The statement given is that $\lim _ { x \rightarrow \theta ^ { - } } f ( x ) = L$ and $\lim _ { x \rightarrow \theta ^ { + } } f ( x ) = L _ { r }$, with $L _ { 1 } = L _ { r }$ (presumably a typo, and it should read $L = L _ { r }$). If $L = L _ { r }$, then the limit as $x$ approaches $\theta$ from both sides is the same, and thus the limit exists. Statement III is false because it contradicts this condition by suggesting the limit does not exist.

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

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