The problem you have provided is asking to evaluate the following limit:

$\lim_{x \to 5} \frac{x^2 - 10x + 25}{x^2 - 5}$

Step-by-step solution:

1. Simplify the numerator: The numerator is a quadratic expression $x^2 - 10x + 25$, which can be factored as: $x^2 - 10x + 25 = (x - 5)^2$

2. Denominator analysis: The denominator is $x^2 - 5$, which cannot be factored further directly, but it is already simplified.

Thus, the expression becomes:

$\lim_{x \to 5} \frac{(x - 5)^2}{x^2 - 5}$

3. Direct substitution: If we try to directly substitute $x = 5$ into the expression, the numerator becomes $(5 - 5)^2 = 0$ and the denominator becomes $5^2 - 5 = 25 - 5 = 20$.

Hence, we have:

$\frac{0}{20} = 0$

Conclusion:

The value of the limit is $\boxed{0}$.

Would you like further details or clarifications on any step?

Here are five related questions you might find useful:

1. What is the general process for factoring quadratic expressions?
2. How do you handle limits that result in indeterminate forms like $0/0$?
3. How can you apply L'Hopital's Rule to limits involving fractions?
4. What are the common techniques to simplify complex limits?
5. Can all limits be evaluated by direct substitution?

Tip: Always check if direct substitution is possible before attempting more complex limit-solving techniques like factoring or L'Hopital's Rule.