To discuss the continuity of the function [tex]\( f(x) = |x| \)[/tex] at [tex]\( x = 0 \)[/tex], we need to analyze the behavior of the function around this point. Specifically, we need to check if the limit of [tex]\( f(x) \)[/tex] as [tex]\( x \)[/tex] approaches 0 exists and is equal to the value of the function at [tex]\( x = 0 \)[/tex].
Let's break this down step by step:
1. Left-Hand Limit as [tex]\( x \)[/tex] Approaches 0:
We need to find the limit of [tex]\( f(x) \)[/tex] as [tex]\( x \)[/tex] approaches 0 from the left (that is, [tex]\( x \to 0^- \)[/tex]).
[tex]\[
\lim_{x \to 0^-} f(x) = \lim_{x \to 0^-} |x| = |0.0001| = 0.0001
\][/tex]
2. Right-Hand Limit as [tex]\( x \)[/tex] Approaches 0:
Next, we need to find the limit of [tex]\( f(x) \)[/tex] as [tex]\( x \)[/tex] approaches 0 from the right (that is, [tex]\( x \to 0^+ \)[/tex]).
[tex]\[
\lim_{x \to 0^+} f(x) = \lim_{x \to 0^+} |x| = |0.0001| = 0.0001
\][/tex]
3. Value of the Function at [tex]\( x = 0 \)[/tex]:
We now need to evaluate the function at [tex]\( x = 0 \)[/tex].
[tex]\[
f(0) = |0| = 0
\][/tex]
4. Compare the Limits and Value:
For the function to be continuous at [tex]\( x = 0 \)[/tex], the left-hand limit, the right-hand limit, and the value of the function at [tex]\( x = 0 \)[/tex] must all be equal. We have:
[tex]\[
\lim_{x \to 0^-} |x| = \lim_{x \to 0^+} |x| = 0.0001
\][/tex]
[tex]\[
f(0) = 0
\][/tex]
Since the left-hand limit [tex]\( 0.0001 \)[/tex], the right-hand limit [tex]\( 0.0001 \)[/tex], and the value of the function at [tex]\( x = 0 \)[/tex] [tex]\( 0 \)[/tex] are not all equal (we need [tex]\( 0 = 0.0001 \)[/tex]), the function is not continuous at [tex]\( x = 0 \)[/tex].
Therefore, we conclude that the function [tex]\( f(x) = |x| \)[/tex] is not continuous at [tex]\( x = 0 \)[/tex].
