The expression provided in the image is:

$\frac{2^n + 5^n}{3^n}$

To find the limit of this sequence as $n$ approaches infinity ($n \to \infty$), let's analyze it step by step.

Step 1: Factor Out the Dominant Term

In this sequence, the dominant term in the numerator is $5^n$ because it grows faster than $2^n$ as $n$ increases. So, let's factor out $5^n$ from the numerator:

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

This simplifies to:

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

Step 2: Simplify the Expression

The expression can be further simplified by separating the exponential terms:

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

Step 3: Evaluate the Limit

As $n$ approaches infinity:

• $\left(\frac{2}{5}\right)^n$ approaches 0 because $\frac{2}{5} < 1$.
• $\left(\frac{5}{3}\right)^n$ grows without bound because $\frac{5}{3} > 1$.

So the expression becomes dominated by the term $\left(\frac{5}{3}\right)^n$ and approaches infinity:

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

Conclusion:

The limit of the sequence as $n$ approaches infinity is $\infty$. This means that the sequence diverges to infinity.

Do you need further details or have any other questions?

Here are five related questions to expand on this topic:

1. How does the rate of growth of exponential functions affect the limit of sequences?
2. What would happen if the base of the exponential terms in the numerator and denominator were the same?
3. Can you compare the growth rates of different exponential functions, such as $2^n$ versus $3^n$?
4. How does the concept of dominant terms help in evaluating limits of sequences?
5. What is the significance of identifying whether a sequence diverges or converges?

Tip: When dealing with exponential sequences, always identify the dominant term, as it dictates the behavior of the sequence as $n$ increases.