To determine why the limit does not exist for the expression [tex]\(\lim _{x \rightarrow 0} \frac{x}{|x|}\)[/tex], let's analyze the behavior of the function as [tex]\(x\)[/tex] approaches [tex]\(0\)[/tex] from the left and the right.
### Analyzing the Behavior:
1. As [tex]\(x\)[/tex] approaches 0 from the left [tex]\((x \rightarrow 0^-)\)[/tex]:
- When [tex]\(x\)[/tex] is slightly negative, [tex]\(|x| = -x\)[/tex].
- Therefore, [tex]\(\frac{x}{|x|} = \frac{x}{-x} = -1\)[/tex].
- As [tex]\(x\)[/tex] approaches [tex]\(0\)[/tex] from the left, [tex]\(\frac{x}{|x|}\)[/tex] approaches [tex]\(-1\)[/tex].
2. As [tex]\(x\)[/tex] approaches 0 from the right [tex]\((x \rightarrow 0^+)\)[/tex]:
- When [tex]\(x\)[/tex] is slightly positive, [tex]\(|x| = x\)[/tex].
- Therefore, [tex]\(\frac{x}{|x|} = \frac{x}{x} = 1\)[/tex].
- As [tex]\(x\)[/tex] approaches [tex]\(0\)[/tex] from the right, [tex]\(\frac{x}{|x|}\)[/tex] approaches [tex]\(1\)[/tex].
### Conclusion:
Since the values that [tex]\(\frac{x}{|x|}\)[/tex] approaches are different from the left and the right, there is no single number [tex]\(L\)[/tex] that the function values get arbitrarily close to as [tex]\(x\)[/tex] approaches [tex]\(0\)[/tex].
Hence, the correct choice is:
A. As [tex]\(x\)[/tex] approaches 0 from the left, [tex]\(\frac{x}{|x|}\)[/tex] approaches [tex]\(-1\)[/tex]. As [tex]\(x\)[/tex] approaches 0 from the right, [tex]\(\frac{x}{|x|}\)[/tex] approaches [tex]\(1\)[/tex]. This means there is no single number [tex]\(L\)[/tex] that the function values get arbitrarily close to as [tex]\(x \rightarrow 0\)[/tex].
