Need help with Asymptotic Notation / Ratio Limit Theorem / L'Hopitals Rule.

Please show work and provide detailed explanations for the following exercise. Thank you very much.

Using the ratio limit theorem (Proposition 3.1.4) and (possibly) L'Hopital's rule (Definition 3.17), prove each of the following: a. (n^3 + n)/(2n + lnn) elementof theta (n^2) b. O(n^4) subset O(3") c. n2" + n^^9 elementof O(e^n) d. n^1/2 elementof ohm ((log n)^4) e. n^log_2 n elementof O(2^n)