https://file.aiquickdraw.com/mathbot/file/17272029616631d5hjp7h.jpg

The expression in the image is:

\[
\lim_{n \to \infty} (3^n - 4^n)
\]

### Solution:

As \(n\) approaches infinity, the term \(4^n\) grows much faster than \(3^n\), since the base 4 is larger than the base 3. Therefore, the term \(3^n\) becomes negligible in comparison to \(4^n\), and the expression \(3^n - 4^n\) will be dominated by \(-4^n\) for large \(n\).

Hence, the limit is:

\[
\lim_{n \to \infty} (3^n - 4^n) = -\infty
\]

### Final Answer:
\[
\boxed{-\infty}
\]

Would you like more details or have any questions? Here are some related questions to consider:

1. What is the behavior of exponential functions with different bases as \(n\) approaches infinity?
2. How does the growth of exponential functions compare with polynomial or logarithmic functions?
3. How would this limit change if the bases of the powers were the same, e.g., \(3^n - 3^n\)?
4. What happens if both bases are less than 1, say \(0.5^n - 0.8^n\)?
5. How can limits involving more complex exponential expressions be evaluated?

**Tip:** When comparing exponential functions with different bases, the function with the larger base will dominate as \(n \to \infty\).

lim (3^n - 4^n) as n approaches infinity.

Learn how to solve the limit of the expression (3^n - 4^n) as n approaches infinity. This problem covers key concepts in limits and exponential growth, highlighting how the larger base dominates as n increases. Suitable for students in Grades 10-12.

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