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Let's solve the limit problem step by step.

The given limit is:

\[
\lim_{{n \to \infty}} \frac{4 \cdot 3^n - 3 \cdot 4^{n-1}}{4 \cdot 5^{n+1} + 4 \cdot 3^{n+2}}
\]

### Step 1: Simplify the expression

#### Numerator:
\[
4 \cdot 3^n - 3 \cdot 4^{n-1}
= 4 \cdot 3^n - 3 \cdot \frac{4^n}{4}
= 4 \cdot 3^n - \frac{3}{4} \cdot 4^n
\]

#### Denominator:
\[
4 \cdot 5^{n+1} + 4 \cdot 3^{n+2}
= 4 \cdot 5 \cdot 5^n + 4 \cdot 3^2 \cdot 3^n
= 20 \cdot 5^n + 36 \cdot 3^n
\]

Now the limit becomes:
\[
\lim_{{n \to \infty}} \frac{4 \cdot 3^n - \frac{3}{4} \cdot 4^n}{20 \cdot 5^n + 36 \cdot 3^n}
\]

### Step 2: Identify dominant terms as \(n \to \infty\)
- In the numerator, the term \( \frac{3}{4} \cdot 4^n \) grows much faster than \( 4 \cdot 3^n \) since \( 4^n \) grows exponentially faster than \( 3^n \).
- In the denominator, the term \( 20 \cdot 5^n \) grows faster than \( 36 \cdot 3^n \) since \( 5^n \) grows faster than \( 3^n \).

Thus, the dominant term in the numerator is \( -\frac{3}{4} \cdot 4^n \), and in the denominator, it is \( 20 \cdot 5^n \).

### Step 3: Simplify the limit using dominant terms
Now the limit reduces to:
\[
\lim_{{n \to \infty}} \frac{-\frac{3}{4} \cdot 4^n}{20 \cdot 5^n}
\]

We can cancel out the exponential terms:
\[
= \lim_{{n \to \infty}} \frac{-\frac{3}{4}}{20} \cdot \left(\frac{4}{5}\right)^n
\]

Since \( \left(\frac{4}{5}\right)^n \to 0 \) as \(n \to \infty\), the entire expression tends to 0.

### Final Answer:
\[
\boxed{0}
\]

Would you like a further breakdown of any steps?

### Follow-up Questions:
1. How does the growth of exponential terms affect limits as \(n \to \infty\)?
2. Why is \( \frac{4^n}{5^n} \to 0 \) as \(n \to \infty\)?
3. What role does dominance of terms play in evaluating limits?
4. Can you explain why the smaller exponential terms were discarded?
5. How would the result change if both exponentials had the same base?

### Tip:
When solving limits involving exponential functions, always identify which term grows faster to simplify the problem.

lim (n -> ∞) [ (4 * 3^n - 3 * 4^(n-1)) / (4 * 5^(n+1) + 4 * 3^(n+2)) ]

Explore how to solve a limit problem involving exponential terms like 3^n, 4^n, and 5^n as n approaches infinity. This problem focuses on the asymptotic behavior and dominance of terms in limits. Suitable for undergraduate mathematics students.

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