QuestionDetermine the truth of these statements for all $c$ and $L$: I. $\lim _{x \rightarrow c} f(x)=L \Rightarrow \lim _{x \rightarrow c}|f(x)|=|L|$; II. $\lim _{x \rightarrow c}|f(x)|=|L| \Rightarrow \lim _{x \rightarrow c} f(x)=L$.

Studdy Solution
The second statement is not necessarily true. This is because the limit of the absolute value of a function as $x$ approaches a certain value does not necessarily equal the limit of the function itself as $x$ approaches that value.For example, consider the function $f(x) = -1$ for all $x$. Then, $\lim_{x \rightarrow c} |f(x)| =1$ for all $c$, but $\lim_{x \rightarrow c} f(x) = -1$ for all $c$. So, even though $|| =1$, $$ is not equal to $-1$.
So, the first statement is true and the second statement is false.