For the function $f(x) = 2\left( \frac{1}{x-4} \right) + 3$:

1. Vertical Asymptote occurs where the denominator of the rational function equals zero.

   • Set $x - 4 = 0$, so the vertical asymptote is at $x = 4$.

2. Horizontal Asymptote is determined by examining the behavior as $x$ approaches infinity or negative infinity. Since the function has a rational form with the degree of the numerator being less than the degree of the denominator, the horizontal asymptote will be at $y = 3$ (the constant added to the function).

Thus, the asymptotes are:

• Vertical: $x = 4$
• Horizontal: $y = 3$

Would you like a deeper explanation or any clarification? Here are some related questions:

1. How do we determine the horizontal asymptote of a rational function?
2. What happens to the graph near vertical asymptotes?
3. How would you find the domain of this function?
4. How would you sketch the graph of $f(x)$?
5. What is the behavior of $f(x)$ as $x$ approaches large positive or negative values?

Tip: When dealing with rational functions, always check both the vertical and horizontal asymptotes as part of analyzing their graphs.