00:01
Since there's a bunch of problems, let me just get to the theme in this.
00:04
Whether you're doing positive infinity or negative infinity, this isn't really good notation.
00:11
I'll just write it like that, positive or negative infinity of 1 over x to any power, it will equal 0.
00:19
And the thought process there is whether you're looking at 1 over x or if you're doing, that's if n is odd, or if n is even, it just looks like this.
00:34
But look at the end behavior.
00:35
They're both approaching zero on both ends.
00:39
So how can we apply that in a problem like 28? is the limit as x approaches negative infinity of cosine of 1 over x over 2 plus 1 over x is in both the top and the bottom, this piece and this piece will approach 0.
00:58
So what you're really looking at now is cosine of 0 over 2 plus 0.
01:05
And i don't know if you need the unit circle.
01:08
On the unit circle, cosine of 0 is 1.
01:12
So this answer will equal 1 over 2 or 1 half.
01:18
And a lot of the theme on this problem is based in that.
01:22
As i'm looking at the next one, again, i'm trying to speed up because there's a lot of questions here.
01:27
4 over x cubed minus cosine of 1 over x and then 8 plus cosine of 1 over x is this piece equal 0 this piece will equal 0 and it doesn't matter if it's negative or positive infinity so what we're looking at is 0 minus cosine of 0 i've already evaluated for you as 1 and then same thing on this one 8 plus 1 because again this is really 0 cosine of 0 is 1.
01:58
And negative 1 times 9 is equal to negative 9.
02:04
So moving on to the next one, let me just give you the answer.
02:10
The answer is 0 on 30, and the reason is because what you can actually do is rewrite this and divide each piece by x squared.
02:34
Kind of bear with me here.
02:43
Because three of them will reduce to something over x or 3 over x squared.
02:50
Like we get a bunch of zeros.
02:52
Of the remaining pieces, though, we get 1 plus 0 over x plus 7.
02:58
But we have another situation where we have 1 over x, which is going to approach 0 for the same reason i had earlier.
03:06
It doesn't matter if we shift the graph left or right.
03:09
It's still going to approach 0 as we go to infinity...