I've placed this on the Math Stack Exchange even though it is really a CS question because it is the math that is stumping me.
Please note, I'm not asking you to do this problem for me, just to make sure I'm not wasting my life on an impossible problem
For the function $t(n)=\sqrt{10n^2 + 7n + 3}$, indicate the class $\Theta(g(n))$ that the function belongs to. (Use the simplest $g(n)$ possible in your answers.) Prove your assertions.
For $ n\ge 1$, this function exhibits linear growth and can be bounded by functions $c_2 g(n)=1n$ and $c_1 g(n)=5n$, so it belongs to $\Theta(n)$. I've checked my intuition on this computationally and I am satisfied that this is the case.
My problem is that I cannot justify this. I've tried factoring the polynomial portion without any luck (no real roots, apparently?). When I try to set up $\lim_{n\to\infty} \frac{t(n)}{g(n)}$, I find I can neither simplify it nor achieve an answer via repeated applications of L'Hopital's Rule. I am at wits' end here. Am I missing something, or does this problem belong on the errata page for the textbook in question?
PS: Textbook in question is Introduction to Design and Analysis of Algorithms, 2nd Edition by Anany Levitin. This is problem 2.2.3. My google-fu has not yielded the errata page for this textbook.