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!