Relation between two asymptotics

Question

What is is the largest $0<c<4$ that satisfies $$2^{n + \sqrt{n}} \in O((4-c)^n)$$ ?

Answer 1

For any fixed $c\in(-\infty, 4)$, we have $$ \frac{2^{n+\sqrt{n}}}{(4-c)^n} = 2^{n+\sqrt{n} - n\log_2 (4-c)} = 2^{(1-\log_2 (4-c))n + \sqrt{n}} \xrightarrow[n\to\infty]{} \begin{cases} \infty & \text{ if } 1-\log_2 (4-c) \geq 0 \\ 0 & \text{ if } 1-\log_2 (4-c) < 0 \\ \end{cases} $$ so for your satement to hold, you need (and it is sufficient) $1-\log_2 (4-c) < 0$, or equivalently $c < 2$.