Question

question 1 says I have to use squeeze
question 2 just approximation whats the difference? details to follow...


side question is there a way to get the equation editor in the first question box?

Answer 1

1. \[a(n) = (\sin (3n)) / (1+3n^2)\]

2. \[a(n) = n/ (2n + \ln n)\]

Answer 2

and the question is: are they convergent? and if so, find the limit

Answer 3

I assume those are sequences? If so, the difference is this: sin is a periodic function, that is, it will stay around some limit for a certain period of time; in other words, sin is always in between [-1,1]. That way, you can "squeeze" it in between two functions g and f such that g <= sin <= f.
For the second part, you can compare it with n/2n. Clearly, n/(2n + ln n) is smaller, because the denominator is larger, and n/2n converges, so, n/(2n + ln n) also converges.

Answer 4

Also, for the second limit, use L'Hopital rule, it should be somewhat easy.

Answer 5

ok cool i think i get it

Answer 6

I will try to make an allusion that will, very likely fail, but here it goes: sin is never reaching a limit, but it's bounded. It will be forever between [-1,1], so we can make a "sandwich" out of it. n, in counterpart, is unbounded and unlimited, meaning that it will go to infinity. We can never fit a large enough "sandwich" to fit it in. Yeah, I know, bad idea to say stuff like these, haha.

Answer 7

sandwich works like squeeze for me thanks