The given mathematical expression is:

\lim_{x \to -2} \frac{\frac{\frac{1}{x} + \frac{1}{2}}{x + 2}}

Let’s solve this step by step:

Step 1: Simplify the numerator

The numerator of the main fraction is:

$\frac{1}{x} + \frac{1}{2}$

This can be written as:

$\frac{2 + x}{2x}$

Step 2: Substitute the numerator into the fraction

The overall expression becomes:

\lim_{x \to -2} \frac{\frac{\frac{2 + x}{2x}}{x + 2}}

Step 3: Combine the nested fraction

Simplify the inner fraction $\frac{\frac{2 + x}{2x}}{x + 2}$:

$\frac{\frac{2 + x}{2x}}{x + 2} = \frac{2 + x}{2x(x + 2)}$

So the overall expression is:

$\lim_{x \to -2} \frac{2 + x}{2x(x + 2)}$

Step 4: Cancel terms

The term $2 + x$ in the numerator cancels with $2 + x$ in the denominator, leaving:

$\lim_{x \to -2} \frac{1}{2x}$

Step 5: Substitution

Substitute $x = -2$ into $\frac{1}{2x}$:

$\frac{1}{2(-2)} = \frac{1}{-4} = -\frac{1}{4}$

Final Answer:

$\boxed{-\frac{1}{4}}$

Do you have any questions or need additional clarification? Here are some related questions:

1. How does substituting values directly help in evaluating limits?
2. What should you do if the limit results in an indeterminate form like $\frac{0}{0}$?
3. How can L'Hôpital's Rule be used in similar problems?
4. Why is factorization important in simplifying limits?
5. How does this example demonstrate the use of fraction manipulation in calculus?

Tip: Always simplify expressions before substituting values to avoid unnecessary complexity.