Let's solve the given limits one by one.

1. Limit 1

$\lim_{n \to \infty} \frac{n^2 - n}{(n + 1)!}$

Solution: For large $n$, the factorial $(n+1)!$ grows much faster than any polynomial expression like $n^2 - n$. Therefore, this limit approaches 0 as $n \to \infty$.

$\lim_{n \to \infty} \frac{n^2 - n}{(n + 1)!} = 0$

2. Limit 2

$\lim_{n \to \infty} \frac{2 \cdot 3^n - 4^n}{4^{n+2} - 3^n}$

Solution: Divide both the numerator and the denominator by $4^n$ to simplify:

$\lim_{n \to \infty} \frac{2 \cdot \left(\frac{3}{4}\right)^n - 1}{4^2 \cdot \frac{1}{4^n} - \left(\frac{3}{4}\right)^n}$

As $n$ approaches infinity, $\left(\frac{3}{4}\right)^n$ approaches 0. So, the expression simplifies to:

$\lim_{n \to \infty} \frac{-1}{16} = -\frac{1}{16}$

3. Limit 3

$\lim_{n \to \infty} \frac{(n - 1)! - (n + 1)!}{(n + 1)! - (n + 2)!}$

Solution: Let's rewrite each factorial term:

1. $(n - 1)!$
2. $(n + 1)! = (n + 1) \cdot n \cdot (n - 1)!$
3. $(n + 2)! = (n + 2) \cdot (n + 1)!$

Substitute these into the limit:

$\lim_{n \to \infty} \frac{(n - 1)! - (n + 1) \cdot n \cdot (n - 1)!}{(n + 1)! - (n + 2) \cdot (n + 1)!}$

After simplifying, this reduces to 0 because both the numerator and denominator become dominated by the factorial terms.

$\lim_{n \to \infty} = 0$

Summary of Results:

1. Limit 1: $\lim_{n \to \infty} \frac{n^2 - n}{(n + 1)!} = 0$
2. Limit 2: $\lim_{n \to \infty} \frac{2 \cdot 3^n - 4^n}{4^{n+2} - 3^n} = -\frac{1}{16}$
3. Limit 3: $\lim_{n \to \infty} \frac{(n - 1)! - (n + 1)!}{(n + 1)! - (n + 2)!} = 0$

Would you like further details on any specific limit or have additional questions?

Related Questions:

1. How does the growth of factorials compare to exponential and polynomial functions?
2. What are some techniques for evaluating limits involving factorials?
3. Can L'Hospital's Rule be applied to any of these limits? Why or why not?
4. How can we use the ratio test to determine the convergence of series involving factorials?
5. What is Stirling's approximation, and how does it help in analyzing factorial limits?

Tip: For limits involving factorials, always compare growth rates. Factorials grow faster than exponential functions, which in turn grow faster than polynomials.