Yes, you can find the best rational approximation of a real number $X$ using continued fractions, particularly when you're limited to rational numbers with denominators not exceeding a given integer.

Steps to Find the Best Rational Approximation:

1. Compute the Continued Fraction Expansion: The first step is to find the continued fraction representation of $X$. This involves expressing $X$ as a sequence of integer parts and fractional remainders.

2. Construct the Approximations: From the continued fraction, you can derive the convergents, which are the best rational approximations. The $n$-th convergent is derived from the first $n$ terms of the continued fraction.

3. Limit the Denominator: As you generate the convergents, you can filter them based on your restriction on the denominator (i.e., $\leq k$).

4. Select the Best Approximation: Among the filtered convergents, select the one that best approximates $X$.

Example:

If $X = \pi$ and you want rational approximations with a denominator $\leq 7$:

1. The continued fraction expansion of $\pi$ is approximately $[3; 7, 15, 1, 292, \ldots]$.
2. The convergents are $\frac{22}{7}$, $\frac{333}{106}$, etc.
3. Here, $\frac{22}{7}$ has a denominator of 7 and is a good approximation of $\pi$.

If you have a specific number $X$ and a maximum denominator, I can help you find the best rational approximation! Would you like to provide a specific number?

Further Questions:

1. What real number do you want to approximate?
2. What is the maximum denominator you want to consider?
3. Are you familiar with how to compute continued fractions?
4. Would you like to see a detailed example with calculations?
5. Do you want to know about applications of rational approximations?

Tip:

For any real number, continued fractions often provide much better approximations than simple fractions, especially with limited denominators.