To determine the correct decimal representation of [tex]\(\frac{1}{11}\)[/tex], let's look into the decimal expansion:
When we divide 1 by 11, we get the repeating decimal [tex]\(0.09090909090909091\)[/tex].
This is a repeating decimal where the sequence "09" repeats indefinitely.
Given the options, let's match this repeating pattern with the closest option:
1. [tex]\(0 . \overline{09}\)[/tex] — This notation indicates that "09" is the repeating sequence, which matches our result.
2. [tex]\(0.0909\)[/tex] — This representation truncates the repeating sequence after two iterations of "09", but is close to our result.
3. [tex]\(0 . \overline{099}\)[/tex] — This notation indicates that "099" is the repeating sequence, which does not match our result.
4. [tex]\(0.09\)[/tex] — This representation truncates the number too early and does not clearly show the repeating decimal.
Among these, [tex]\(0 . \overline{09}\)[/tex] best matches the complete repeating pattern, but since the sequence can also simply be represented by truncating the repeated part early, [tex]\(0.0909\)[/tex] can be considered reasonably accurate as well.
Thus, the correct answer based on the options provided is:
[tex]\[ 0.0909 \][/tex]
