Question

Question #57453

Answer 1

See explanation section below.

Explanation:

In this case,

`abs(a_n-L) = abs((2sinn)/n - 0) = abs((2sinn)/n)`.

For positive `n`, this is `= 2/nabs(sin n)`

Recall that for all real `n`, we have `abs(sin n) <= 1`

So for all `n > 0`, we know that

`abs((2sinn)/n) = 2/nabs(sin n) <= 2/n`.

By making `M` sufficiently great, we can make `2/n < epsilon` for ann `n >= M`.

(Take `M` to be any number greater than `2/epsilon`.)

Answer 2

Very very simply

Explanation:

We already must know that `sin x` is between -1 and +1 for every real x, so that `-1 <= sin2n <=1` for every n (or indeed any real number). When you divide that by n as `n->∞`, then it is clear that the answer is zero. "Everything should be made as simple as possible, although not any simpler"