Intuitively I know that $ 5^{xf(x)} \in \mathcal O (5^{f(x)}) $ but how would I go about proving this? I am at my wit's end on this.
$ 5^{xf(x)} \in \mathcal O (5^{f(x)}) \Leftrightarrow \exists c, B \in \mathbb{R}^+, \forall n \in \mathbb{N}, n \geq B \Rightarrow 5^{xf(x)} \leq c * 5^{f(x)} $
Is it really necessary to pin down a specific $ c $ and $ B $? How would I go about acquiring a general solution for this?
Thanks.
$$ \frac{5^{x f(x)}}{5^{f(x)}} = 5^{(x-1)f(x)} $$
This probably goes to infinity, which implies
$$ 5^{x f(x)} = \omega(5^{f(x)})$$
although if your $f$ was chosen so that the limit goes to $0$, it would be little-oh. Or if the limit went to a finite number, it would be $\Theta$