Prove or disprove that 𝑓(𝑛) ∈ 𝑜(𝑔(𝑛)) where 𝑓(𝑛) = 𝑛1.5 + log2
𝑛 and 𝑔(𝑛) = 𝑛2
.

To prove or disprove that \( f(n) \in o(g(n)) \) for the given functions:

- \( f(n) = n^{1.5} + \log_2 n \)
- \( g(n) = n^2 \)

we need to determine whether \( f(n) \) is asymptotically smaller than \( g(n) \), which means that:

\[
\lim_{{n \to \infty}} \frac{f(n)}{g(n)} = 0
\]

### Step-by-Step Analysis

1. **Define the ratio \( \frac{f(n)}{g(n)} \):**

\[
\frac{f(n)}{g(n)} = \frac{n^{1.5} + \log_2 n}{n^2}
\]

2. **Simplify the expression:**

\[
\frac{f(n)}{g(n)} = \frac{n^{1.5}}{n^2} + \frac{\log_2 n}{n^2}
\]

\[
= \frac{1}{n^{0.5}} + \frac{\log_2 n}{n^2}
\]

3. **Analyze the limit as \( n \to \infty \):**

We need to evaluate:

\[
\lim_{{n \to \infty}} \left( \frac{1}{n^{0.5}} + \frac{\log_2 n}{n^2} \right)
\]

- Consider the first term: 

\[
\lim_{{n \to \infty}} \frac{1}{n^{0.5}} = 0
\]

- Consider the second term:

\[
\lim_{{n \to \infty}} \frac{\log_2 n}{n^2}
\]

To compute this, notice that \( \log_2 n \) grows much more slowly than \( n^2 \). Thus, we apply L'Hopital's Rule:

\[
\lim_{{n \to \infty}} \frac{\log_2 n}{n^2} = \lim_{{n \to \infty}} \frac{\frac{d}{dn} (\log_2 n)}{\frac{d}{dn} (n^2)} = \lim_{{n \to \infty}} \frac{\frac{1}{n \ln 2}}{2n} = \lim_{{n \to \infty}} \frac{1}{2n^2 \ln 2} = 0
\]

4. **Combine the limits:**

\[
\lim_{{n \to \infty}} \left( \frac{1}{n^{0.5}} + \frac{\log_2 n}{n^2} \right) = 0 + 0 = 0
\]

### Conclusion

Since the limit is 0, we have:

\[
f(n) \in o(g(n))
\]

Therefore, **the statement is true**: \( f(n) = n^{1.5} + \log_2 n \in o(n^2) \).

### Would you like more details or have any questions?

Here are 5 related questions to consider:
1. What is the definition of Big-O notation and how does it differ from Little-o?
2. How can L'Hopital's Rule be applied in other asymptotic analyses?
3. What are examples of other functions \( f(n) \) and \( g(n) \) where \( f(n) \in o(g(n)) \)?
4. How does the logarithmic function compare to polynomial functions in terms of growth rate?
5. What are some applications of asymptotic analysis in computer science?

**Tip:** Remember, \( f(n) \in o(g(n)) \) implies that \( f(n) \) grows strictly slower than \( g(n) \) as \( n \to \infty \).

Prove or disprove that 𝑓(𝑛) ∈ 𝑜(𝑔(𝑛)) where 𝑓(𝑛) = 𝑛^1.5 + log2 𝑛 and 𝑔(𝑛) = 𝑛^2.

Explore the step-by-step solution to proving or disproving that f(n) = n^1.5 + log2(n) belongs to o(n^2). This problem involves asymptotic analysis, limits, and the application of L'Hopital's Rule, making it ideal for undergraduate students studying mathematics or computer science.

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