Section 7.4 Asymptotic Equality

Make sure you read the questions very carefully and see on what its being asked for. Please be careful with this assignment.

MATH 1522: SECTION 7.4 WRITTEN ASSIGNMENT Please read the reading assignment associated with this section before doing this assignment. The purpose of this assignment is to apply the methods learned in that reading to compute limits. Using a different method will result in no credit. 1. Use asymptotic equality to compute the limit lim (3n + 4) (n - 2) 1-+00 (2n + 3) (2n + 1) Write out the asymptotics that you are using separately. 2. Use asymptotic equality to compute the limit 2n+(n + 3) In(n + 1) lim n-100 2" (3n + 1) In(n) Write out the asymptotics that you are using separately. It might be worth simplifying before using asymptotics in this example. 3. Show the asymptotic equality In(n2 + 1) ~ 2In(n) as n - co by using the definition of asymptotic equality. There is both an example of this in the reading and a video example of this, so please consult those if you're having trouble.Section 7.4 Reading Assignment: Relative Rates of Growth1 Instructions. Read through this assignment and complete the three exercises below based on this reading. In Chapter 10, we will encounter lots of different limits as n - co. Fortunately for us, limits at infinity will be the only ones we're concerned with in this chapter, but there is a lot going on with these limits and some intuition that it'd be good to develop about these limits. Hopefully you remember L'Hopital's rule and the fact that - is an indeterminate form, meaning that its value depends on the particulars of the problem. This illustrates that the use of the infinity symbol as a single quantity actually removes a lot of information about a limit. To take an example, while both 3n and n' approach co (as n - co), n is, in a sense, faster than 3n. This can be seen when taking the limit of the fraction lim 3n = lim - = 0. This is what I meant before: there are many different speeds that n+00 n2 n+00 n we can approach infinity, so - can have any value (including infinity). In lim = = 0, the expressions 3n and n' are "racing" with each other: 3n in the numerator is pulling the limit closer to infinity, while n in the denominator is pulling the limit closer to zero. The result shows n completely won that race. In other cases, there's something like a tie in the race, which we will explore later. We take this as an inspiration for defining what we mean by one sequence growing faster than another: Definition. A sequence an is growing slower than another sequence by which is written an