You have the function:

$y = \frac{\log(x)}{x}$

This is a mathematical expression where the logarithm of $x$ is divided by $x$.

To analyze this function:

1. Domain: The logarithm $\log(x)$ is only defined for $x > 0$, so the domain of the function is $x > 0$.

2. Asymptotes:

   • As $x \to 0^+$, $\log(x)$ tends to $-\infty$, and the fraction $\frac{\log(x)}{x}$ goes to $-\infty$. Hence, there is no vertical asymptote at $x = 0$, but the function tends to negative infinity as $x$ approaches zero from the right.
   • As $x \to \infty$, $\log(x)$ grows slower than $x$, and thus $\frac{\log(x)}{x} \to 0$. So the horizontal asymptote is $y = 0$.

3. Derivative: To find the critical points and behavior of the function, we can differentiate it.

$\frac{d}{dx} \left( \frac{\log(x)}{x} \right) = \frac{x \cdot \frac{d}{dx}(\log(x)) - \log(x) \cdot \frac{d}{dx}(x)}{x^2}$

This simplifies to:

$\frac{1 - \log(x)}{x^2}$

4. Critical Points: To find the critical points, set the derivative equal to zero:

$1 - \log(x) = 0 \implies \log(x) = 1 \implies x = e$

At $x = e$, we have a potential local maximum.

5. Second Derivative Test: To confirm if this is a maximum, you can take the second derivative.

Would you like to explore any specific aspect of this function, such as its graph, the behavior near certain points, or further calculations?