I have the function:
$y = f(x) = \frac{x}{\ln x}$
The function is undefined for the conditions:
- a denominator of a fraction being zero.
- a logarithm being negative or equal to zero.
Hence, is the system:
$$ \begin{cases} & x \ne 0\\ & x > 0 \end{cases} $$
and the domain is $\forall x\in\mathbb R: x > 0$
Following the result showed in wolframalpha the domain is $\forall x\in\mathbb R: x > 1$ or $0 < x < 1$
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- a logarithm being negative.
Watch out: you want the argument of the logarithm to be positive, not the logarithm itself ($\ln x$ will be negative if $0 < x < 1$, that's not a problem).
So the denominator can't be $0$ which means $\ln x$ can't be zero which means $x$ can't be $1$: $$\color{blue}{x \ne 1}$$
Secondly, for $\ln x$ to exist, $x$ has to be strictly positive: $$\color{blue}{x > 0}$$
Combining both conditions yields: $0 < x < 1$ or $x > 1$, or written as a set: $$x \in (0,1) \cup (1,+\infty)$$
Hence, is the system: $$ \begin{cases} & \color{red}{x \ne 0}\\ & x > 0 \end{cases} $$
Your error is in red; see above: it is $x=1$ that makes the denominator $0$, not $x=0$.
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