


8.2 Determine the limits of the following sequences, and then prove your claims. (a) (n =...
Find the limits of the following sequences, if they exist. n+ (-1)" COS n An = an = n
Provide an ? N proof to prove that the following sequences
converge.
Question (e), please.
5. Provide an e – N proof to prove that the following sequences converge. (a) {ne cos(n)} (b) {zo Bom} (c) {(-1)In (n)} (d) an = 2 + 1 (@) an = V1 -
Provide a complete and accurate e-N proof that the following sequences converge. That is, prove these sequences converge. 2Ti3 1
Provide a complete and accurate e-N proof that the following sequences converge. That is, prove these sequences converge. 2Ti3 1
1. (18 points) Determine if the following limits exist. In each case prove and explain your argument. (a) (c) lim *-(1,5) x+y 4.xy2 2+(0,0) x2 + y2 xy lim (b) x?y2 lim x-(0,0) x4 + 3y
=) Determine if the following limits exist. In each case prove and explain your argument. (c) xy lim *+(1,5) x + y 4xy2 lim *-*(0,0) x2 + y2 lim x²y² *+(0,0) x4 + 3y4
1. Determine if the following limits exist. In each case prove and explain your argument. (c) lim x +y + y sin x siny *(0,0) XY lim x-(0,0) x4 + y2 lim x+(0,0) x2y2 + (x + y2)2
1. Determine if the following limits exist. In each case prove and explain your argument. (a) lim x+y + xy sin x siny x²y lim *+(0,0) x4 + y2 x(0,0) xy
for every n. Prove: If (a) converges, then 11. Let (a.) and (b) be sequences such that a, b, < so does (bn). There are several ways to prove this; at least one doesn't involve Cauchy sequences or e. Be careful though you don't know that () converges so make sure that your method of proof doesn't in fact require (b) to converge.
1. (18 points) Determine if the following limits exist. In each case prove and explain your argument. (a) (0) lim ху 4xy? 2+(1,5) x+y lim **(0,0) x2 + y2 (b) x²y² lim *-*(0,0) *4 + 3y 2 (14 neinte) Find the derivative hole
Prove that the following sequences diverge to infinity or
negative infinity
(c) { - e^n }
(a) 120-1)) 3 n2 Hint : Provide an M - N proof that an approaches infinity
(a) 120-1)) 3 n2 Hint : Provide an M - N proof that an approaches infinity