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To evaluate the limit

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

we can analyze the behavior of each term as \( n \) approaches infinity.

### Step-by-Step Solution

1. **Identify Dominant Terms**: As \( n \) becomes very large, the exponential term with the largest base will dominate both in the numerator and the denominator.

   - In the numerator: \( 3^n \) grows faster than \( 2^n \).
   - In the denominator: \( 5^n \) grows faster than \( 2^n \).

2. **Rewrite the Expression**: We can factor out the dominant term in both the numerator and the denominator to simplify the expression.

   \[
   \frac{2^n - 3^n}{5^n + 2^n} = \frac{3^n \left(\frac{2^n}{3^n} - 1\right)}{5^n \left(1 + \frac{2^n}{5^n}\right)}
   \]

3. **Simplify the Fractions Inside the Parentheses**:
   - \(\frac{2^n}{3^n} = \left(\frac{2}{3}\right)^n\), which approaches 0 as \( n \to \infty \) because \( \frac{2}{3} < 1 \).
   - \(\frac{2^n}{5^n} = \left(\frac{2}{5}\right)^n\), which also approaches 0 as \( n \to \infty \) because \( \frac{2}{5} < 1 \).

4. **Take the Limit**: Substitute these results into the expression:

   \[
   \lim_{n \to \infty} \frac{2^n - 3^n}{5^n + 2^n} = \frac{3^n (0 - 1)}{5^n (1 + 0)} = \frac{-3^n}{5^n}
   \]

5. **Further Simplify**: Rewrite \(\frac{-3^n}{5^n}\) as \(-\left(\frac{3}{5}\right)^n\). Since \(\frac{3}{5} < 1\), this fraction raised to the power \( n \) will approach 0 as \( n \to \infty \).

6. **Final Answer**:

   \[
   \lim_{n \to \infty} \frac{2^n - 3^n}{5^n + 2^n} = 0
   \]

### Conclusion
The limit is:
\[
0
\]

Would you like further details or have any questions about this process?

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Here are some additional questions related to this concept:

1. How would the limit change if the numerator and denominator had different dominant terms?
2. What techniques can be used if the terms do not simplify with direct factoring?
3. Can we apply L'Hôpital's Rule to solve limits with exponential functions, and how?
4. How does changing the bases of the exponential functions affect the limit result?
5. What other types of limits approach zero using similar reasoning?

**Tip:** When evaluating limits involving exponential functions, identifying the term with the largest base is crucial, as it dictates the behavior as \( n \) approaches infinity.

Evaluate the limit \( \lim_{{n \to \infty}} \frac{2^n - 3^n}{5^n + 2^n} \).

Understand how to evaluate the limit \( \lim_{{n \to \infty}} \frac{2^n - 3^n}{5^n + 2^n} \) by identifying dominant exponential terms. Learn key concepts in limits, asymptotic behavior, and exponential functions, suitable for high school calculus students.

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