Given \( \lim _{x \rightarrow 0^{-}} f(x)=L_{1}, \lim _{x \rightarrow 0^{+}} f(x)=L_{2} \), and \( L_{1}=L_{2} \), which of the following statements is FAL SE? I. \( \lim _{x \rightarrow 0} f(x)=L_{1} \) II. \( \lim _{x \rightarrow 0} f(x)=L_{2} \) III \( \lim _{x \rightarrow 0} f(x) \) does not exist.

To analyze the statements given the conditions \( \lim _{x \rightarrow 0^{-}} f(x)=L_{1} \), \( \lim _{x \rightarrow 0^{+}} f(x)=L_{2} \), and \( L_{1}=L_{2} \), we can use the definition of the limit. Since both one-sided limits exist and are equal (i.e., \( L_{1} = L_{2} \)), we can conclude that the two-sided limit exists and is equal to that common value. Therefore, we can state: \[ \lim_{x \rightarrow 0} f(x) = L_{1} = L_{2} \] Now, let's evaluate each statement: I. \( \lim _{x \rightarrow 0} f(x)=L_{1} \) This statement is true because we have established that the limit exists and equals \( L_{1} \). II. \( \lim _{x \rightarrow 0} f(x)=L_{2} \) This statement is also true for the same reason; the limit exists and equals \( L_{2} \), which is equal to \( L_{1} \). III. \( \lim _{x \rightarrow 0} f(x) \) does not exist. This statement is false because we have shown that the limit does exist and is equal to \( L_{1} \) (and \( L_{2} \)). Thus, the false statement among the options is: **III. \( \lim _{x \rightarrow 0} f(x) \) does not exist.**