Find Domain of the following: $$ h(x)=\frac{x}{\ln x} $$ A) $$ R $$ B) $$ R-\{0\} $$ C) $$ ] 0, \infty[ $$ D) $$ ] 0,1[\mathrm{U}] 1, \infty[ $$

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To find the domain of the function $$ h(x) = \frac{x}{\ln x} $$, we need to determine the values of $$ x $$ for which the function is defined.

1. **Denominator Condition**: The function has a denominator $$ \ln x $$. The natural logarithm $$ \ln x $$ is defined for $$ x > 0 $$. However, we also need to ensure that $$ \ln x \neq 0 $$ to avoid division by zero. The logarithm $$ \ln x = 0 $$ when $$ x = 1 $$. Therefore, we must exclude $$ x = 1 $$ from the domain.

2. **Combining Conditions**: Since $$ \ln x $$ is defined for $$ x > 0 $$ and we need to exclude $$ x = 1 $$, the domain of $$ h(x) $$ is all positive real numbers except for 1.

Thus, the domain can be expressed as:
$$
(0, 1) \cup (1, \infty)
$$

This corresponds to option D: $$ ] 0,1[ \mathrm{U} ] 1, \infty[ $$.

Therefore, the correct answer is:
$$
\text{D) } ] 0,1[ \mathrm{U} ] 1, \infty[
$$
The domain of $$ h(x) = \frac{x}{\ln x} $$ is $$ ] 0,1[ \mathrm{U} ] 1, \infty[ $$.

Find Domain of the following:
A)
B)
C)
D)

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