The given mathematical expression is:

$\lim_{{x \to 0}} \frac{\frac{1}{4 + x} - \frac{1}{4}}{x}$

This is a limit that resembles the definition of a derivative. Let's break it down and find the limit.

Step-by-step solution:

1. Simplify the numerator: We have $\frac{1}{4 + x} - \frac{1}{4}$ in the numerator. To simplify this, let's get a common denominator: $\frac{1}{4 + x} - \frac{1}{4} = \frac{4 - (4 + x)}{4(4 + x)} = \frac{-x}{4(4 + x)}$

2. Substitute this back into the limit: Now, substitute this into the original expression: $\lim_{{x \to 0}} \frac{\frac{-x}{4(4 + x)}}{x}$

3. Simplify the expression: Cancel $x$ from both the numerator and the denominator: $\lim_{{x \to 0}} \frac{-1}{4(4 + x)}$

4. Evaluate the limit as $x$ approaches 0: When $x \to 0$, the expression becomes: $\frac{-1}{4 \cdot 4} = \frac{-1}{16}$

Thus, the value of the limit is: $\boxed{-\frac{1}{16}}$

Let me know if you would like further clarification on any of the steps!

Follow-up Questions:

1. What is the derivative definition related to this limit?
2. Can this limit be solved using L'Hopital's Rule?
3. How do you simplify complex rational expressions like this limit?
4. What is the significance of finding limits at points of indeterminacy?
5. How does the concept of continuity apply in evaluating limits?

Tip:

When simplifying complex fractions, always check if you can factor or cancel terms before applying more advanced methods like L'Hopital's Rule.