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The given problem is:  

\[
\lim_{n \to \infty} \frac{(n+1)! + (n-1)!}{n!}.
\]  

To evaluate this limit, let’s simplify the expression step by step.

---

### **Step 1: Factor out \( n! \) from the numerator**
We begin by factoring \( n! \) out of each term in the numerator:

\[
(n+1)! + (n-1)! = (n+1) \cdot n! + (n-1)!.
\]

Rewriting the original expression:

\[
\frac{(n+1)! + (n-1)!}{n!} = \frac{(n+1)n! + (n-1)!}{n!}.
\]

Now, divide each term in the numerator by \( n! \):

\[
\frac{(n+1)! + (n-1)!}{n!} = (n+1) + \frac{(n-1)!}{n!}.
\]

---

### **Step 2: Simplify \( \frac{(n-1)!}{n!} \)**
The term \( \frac{(n-1)!}{n!} \) can be written as:

\[
\frac{(n-1)!}{n!} = \frac{1}{n}.
\]

Thus, the expression simplifies to:

\[
\frac{(n+1)! + (n-1)!}{n!} = (n+1) + \frac{1}{n}.
\]

---

### **Step 3: Take the limit as \( n \to \infty \)**
Now take the limit of each term as \( n \to \infty \):

- The first term \( (n+1) \to \infty \),
- The second term \( \frac{1}{n} \to 0 \).

Therefore:

\[
\lim_{n \to \infty} \left( (n+1) + \frac{1}{n} \right) = \infty.
\]

---

### **Final Answer:**
The limit diverges to infinity:

\[
\lim_{n \to \infty} \frac{(n+1)! + (n-1)!}{n!} = \infty.
\]

---

Would you like a detailed explanation of the factorial manipulation or further examples? Let me know!  

Here are 5 related questions to explore:  
1. How do you simplify expressions involving factorials in limits?  
2. What happens if \( (n+1)! \) and \( (n-1)! \) are replaced by powers of \( n \)?  
3. How do factorial growth rates compare to exponential growth rates?  
4. What is the behavior of similar limits where terms include \( n! \) and \( k! \) for \( k < n \)?  
5. How do you apply Stirling's approximation to approximate factorials?  

**Tip:** Always factor out the dominant term when dealing with limits involving factorials!

Evaluate the limit: \( \lim_{n \to \infty} \frac{(n+1)! + (n-1)!}{n!} \).

Learn how to evaluate the limit \( \lim_{n \to \infty} \frac{(n+1)! + (n-1)!}{n!} \) with step-by-step simplifications. This problem involves key concepts like factorial manipulation, asymptotic growth, and limit analysis. Ideal for advanced high school or undergraduate math students.

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