To solve the given limit, let's analyze the problem step-by-step. The expression to solve is:

$\lim_{x \to -5} \frac{x + 5}{\frac{1}{x} + \frac{1}{5}}.$

Step 1: Simplify the denominator

The denominator is: $\frac{1}{x} + \frac{1}{5}.$ Finding a common denominator, we get: $\frac{1}{x} + \frac{1}{5} = \frac{5 + x}{5x}.$

Step 2: Substitute the simplified denominator back

The entire expression becomes: $\lim_{x \to -5} \frac{x + 5}{\frac{5 + x}{5x}}.$ Simplify this fraction by multiplying by the reciprocal of the denominator: $\frac{x + 5}{\frac{5 + x}{5x}} = (x + 5) \cdot \frac{5x}{5 + x}.$

Notice that $x + 5 = 5 + x$, so they cancel out: $(x + 5) \cdot \frac{5x}{5 + x} = 5x.$

Step 3: Take the limit

Now, the simplified expression is: $\lim_{x \to -5} 5x.$ Substitute $x = -5$: $5(-5) = -25.$

Final Answer:

$\boxed{-25}$

Would you like further clarification on this? Let me know!

Related Questions:

1. How do you find the limit of a fraction with complex denominators?
2. How do cancellations help simplify limits?
3. What happens if the numerator and denominator both approach zero?
4. What is L'Hôpital's Rule and how does it apply to limits like this?
5. How do you evaluate a limit as $x \to \infty$ with a similar structure?

Tip: Always simplify fractions before applying the limit to avoid unnecessary complications.