1. Answer the following questions about asymptotic equality. (a) Use asymptotic equality to compute the limit (3x2 + 1) In(2) lim 1-+00 (2x + 1)2 In (5x + 3) Explicitly state the asymptotic equalities used for this computation. Do not use a different method for this problem. (b) Use the definition of asymptotic equality to show that the asymptotic equality In(24 + 2) ~41(2) as x - co is true. Explain how your computation shows this.2. Consider the improper integral 1 (a) Explain why the integral is improper, Include any computations if needed. (b) Rewrite the improper integral using the limit definition of the improper integral. c) Determine whether the improper integral converges or diverges by computing the answer obtained in part (b).3. Consider the following differential equation dy d - 4m3J 121'3 a: with the initial condition y(0) = 5. (a) State whether the dierential equation is linear or separable. If both, then just state the one relevant to the method you will use to solve the equation in part {b}. (b) Solve the initialvalue problem. Show your steps for this process. ((3) Check that your solution is correct. Be sure to Show your work thoroughly for this. 4. Consider the following differential equation xy' + 3y * 203:2 with the initial condition y(1) 3. (a) State whether the differential equation is linear or separable. If both, state the one that relevant to the method you will use to solve the equation in part {13). (b) Solve the initial-value problem. Show your steps for this process. [6) Check that your solution is correct. Be sure to Show your work thoroughly for this. 5. Determine whether each of the series below converges or diverges, explicitly stating your con- clusion. State the test(s) used to determine your answer (only the relevant test(s)) and show the necessary computations for using those tests. 16712 (a) Z 4 ":011 +8 6. Determine whether each of the series below converges absolutely, converges conditionally, or diverges, explicitly stating your conclusion. State the test(s) used to determine your answer (only the relevant test(s)) and show the necessary computation for using those tests. (a) (i)nun 1 4n + 3 (b) (-1)n+In n=] n2+4