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resuelvelo sin formulas como L.Hopital y de manera detallada

To solve the limit \(\lim_{x \to 5} \frac{x^2 - 25}{x^2 - 5x} = 10\) without using L'Hôpital's Rule, we can approach it by simplifying the expression.

### Step 1: Factor the Numerator and Denominator
The expression given is:
\[
\lim_{x \to 5} \frac{x^2 - 25}{x^2 - 5x}
\]
Notice that the numerator \(x^2 - 25\) is a difference of squares and can be factored:
\[
x^2 - 25 = (x - 5)(x + 5)
\]

The denominator \(x^2 - 5x\) can be factored by taking out a common factor:
\[
x^2 - 5x = x(x - 5)
\]

### Step 2: Simplify the Expression
Substituting the factored forms into the expression gives:
\[
\lim_{x \to 5} \frac{(x - 5)(x + 5)}{x(x - 5)}
\]
We can now cancel out the common factor \((x - 5)\) in the numerator and denominator, assuming \(x \neq 5\) (which is valid as we are considering the limit as \(x\) approaches 5, but not at \(x = 5\)):
\[
\lim_{x \to 5} \frac{x + 5}{x}
\]

### Step 3: Evaluate the Limit
Now that the expression is simplified, we can directly substitute \(x = 5\) into the simplified expression:
\[
\frac{5 + 5}{5} = \frac{10}{5} = 2
\]

However, the problem states that the limit equals 10, not 2, which indicates that there is a mistake. Let's clarify that.

### Step 4: Correct Approach
It seems there might have been an error in the problem statement or interpretation. If the problem requires the limit to be 10, then we have to revisit the given expression. Based on the simplifications we've made, the limit evaluates to 2.

If you need the limit to equal 10, please double-check the equation given or confirm any additional steps required. Otherwise, based on the problem statement and standard algebraic methods, the correct limit is:

\[
\lim_{x \to 5} \frac{x^2 - 25}{x^2 - 5x} = 2
\]

Would you like to explore the discrepancy further or require additional details? Here are some related questions:

1. How does the factorization of polynomials help in evaluating limits?
2. What are the common pitfalls when simplifying expressions in limit problems?
3. How does L'Hôpital's Rule differ from direct factorization methods in solving limits?
4. Can there be multiple correct approaches to the same limit problem?
5. What role does substitution play in evaluating limits after simplification?
6. Why is it essential to exclude the point of interest when canceling factors in limits?
7. How can we verify the correctness of a limit after simplification?
8. What are the implications if the limit does not match the expected value in a problem?

**Tip:** When solving limits, ensure to factor and simplify expressions completely before substituting values to avoid errors.

Learn how to solve the limit of (x^2 - 25) / (x^2 - 5x) as x approaches 5 without using L'Hôpital's Rule. Follow a detailed step-by-step solution, including factorization and algebraic simplification techniques. Suitable for high school students and anyone interested in understanding limit problems in algebra.

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