The given expression is:

$\frac{\frac{1}{x} - \frac{x}{3} + \frac{3}{x}}{3}$

To simplify, we will follow these steps:

1. Combine the terms in the numerator:

$\frac{1}{x} + \frac{3}{x} - \frac{x}{3}$

The first two terms, $\frac{1}{x}$ and $\frac{3}{x}$, can be combined as:

$\frac{1 + 3}{x} = \frac{4}{x}$

Now the expression becomes:

$\frac{\frac{4}{x} - \frac{x}{3}}{3}$

2. Find a common denominator for the terms inside the numerator:

To combine $\frac{4}{x}$ and $\frac{x}{3}$, we need a common denominator. The least common denominator (LCD) of $x$ and $3$ is $3x$. So we rewrite each term:

$\frac{4}{x} = \frac{4 \cdot 3}{3x} = \frac{12}{3x}$ $\frac{x}{3} = \frac{x \cdot x}{3x} = \frac{x^2}{3x}$

Now, the numerator becomes:

$\frac{12}{3x} - \frac{x^2}{3x} = \frac{12 - x^2}{3x}$

3. Simplify the whole expression:

Now, the original expression looks like this:

$\frac{\frac{12 - x^2}{3x}}{3}$

Dividing by 3 is equivalent to multiplying by $\frac{1}{3}$:

$\frac{12 - x^2}{3x} \cdot \frac{1}{3} = \frac{12 - x^2}{9x}$

Thus, the fully simplified expression is:

$\frac{12 - x^2}{9x}$

Would you like further clarification on any part of this? Here are some related questions:

1. What are other methods to simplify rational expressions like this?
2. How can we factor the numerator of this expression?
3. How do you solve for $x$ in the simplified equation?
4. What would happen if $x = 0$ in this expression?
5. How do you handle expressions with more complex numerators or denominators?

Tip: When simplifying expressions with fractions, always look for common denominators to combine terms effectively!