I'm given the equation $(n^2+1)^{10}$
I have to find it's efficieny class, and prove it: It looks to be $Big Theta (n^{20})$
I've tried this using limits:
$\lim_{n\to\infty}(n^2+1)^{10}/n^{20} = \lim_{n\to\infty}(n^2+1)^{10}/n^{2(10)} = \lim_{n\to\infty}((n^2+1)/n^2)^{10}$
And looking at the problem, I thought the answer is 0, because when n is infinity, and is plugged in, we're diving a small numerator by a large denominator, which should give zero. And since it's less than $C> 0$, the original equation would not be theta($n^{20}$).
However, after looking at the textbook answer, the answer is actually 1, meaning it is indeed theta($n^{20}$).But I really don't understand why? I'm not sure how that's the case even with the exponent, isn't a small number divided by infinity always going to equal zero?