Worked Examples: Asymptotic Notation

These examples are designed to step through the logical process of applying the algorithmic concepts.

Example 1: Basic Polynomial

❓ Problem: Prove $3n^2 - 100n + 6 \in O(n^2)$.

💡 Solution: Choose $n_0$ such that $-100n$ is overwhelmed... $3n^2 + 6 \le 4n^2$ for $n \ge 3$.

Example 2: Big-Omega

❓ Problem: Prove $n^{1.1} \in \Omega(n)$.

💡 Solution: Use limits or definition. $n^{1.1}/n = n^{0.1} \to \infty$.

Example 3: Limits

❓ Problem: Compare $n \log n$ and $n^{1.5}$.

💡 Solution: Limit is 0. So $n \log n \in o(n^{1.5})$.

Example 4: Little-o

❓ Problem: Is $2n \in o(n)$?

💡 Solution: Limit is 2. Not 0. So False.

Example 5: Exponential

❓ Problem: Is $2^{n+1} \in O(2^n)$?

💡 Solution: Yes. $2^{n+1} = 2 \cdot 2^n$. Constant factor 2.

Example 6: Factorial

❓ Problem: Compare $n!$ and $2^n$.

💡 Solution: $n!$ grows much faster. $n! \in \Omega(2^n)$.