Answer Happy

Please walk me through this in detail and give me any topics to
look to prepared for questions like this, I am trying to study for
the final and I have had trouble understanding this class
Problem 2.1: Find the limits of the following sequences (explain your reasoning) (a) limn+00 m2 21 Solution: Every exponential function with the base greater than 1 grows asymptotically strictly faster than any polynomial functions (per our lecture notes). Therefore, 2n lim = W. n+oo n2 (d) limn+00 n+log3 (n+1) n+1 n n-> Solution: n + log3(n + 1) lim log3(n + 1 lim + lim = 1+0= 1 n+1 noon +1 n-> n+1 The first limit is 1, the second limit is 0 (the latter because logarithms of polynomials in n grow asymptot- ically strictly slower than polynomials in n, as stated in class). 2log3 n (e) limnmo n+1 > n Solution: Observe that 21083 n = nlog3 2 (by properties of logarithms discussed in class). Therefore, = n nlog: 2 2log3 lim n+on+1 lim n+on+1 = 0.