00:02
We're interested in the behavior of this function at the extremes.
00:06
Does it have asymptotes? is there a horizontal? is there a vertical? and how does it behave at those asymptotes if they're there? so first, we're going to reduce this function to its simplest form.
00:20
That means we're going to factor completely.
00:23
The first step, factor out 2x from the numerator.
00:26
That'll leave us an x squared plus 5x plus 6.
00:31
And in the denominator, we can factor out x.
00:35
Squared leaving an x plus two.
00:40
Turns out that this polynomial right here can be factored a little bit further.
00:46
It's actually x plus two times x plus three.
00:55
Now we can reduce.
00:56
We can cancel this x plus two.
00:58
And before we forget about it, we want to make sure we remember that we just now excluded negative two from the domain.
01:05
If we were to substitute in negative two, we'd get a zero in the denominator.
01:08
That would be undefined.
01:11
We can also cancel out one of these xes.
01:14
So this would become x to the first in the denominator.
01:17
And we also just showed that zero is also excluded from the domain.
01:24
Now that we've done that, we have a much, much simplified equation.
01:29
We have two times the quantity x plus three over two over x.
01:40
And that can be simplified just a little bit.
01:43
Further.
01:45
We can separate that fraction, so it's going to be x over x plus three over x and distribute the two.
01:51
So two times x over x is two times one, and two times three over x is six over x.
01:59
So this is now the most reduced form of our function.
02:03
Now what happens as x approaches infinity? as x approaches infinity, 6 over x is a positive number that is shrinking in size approaching zero.
02:18
As this denominator x gets larger and larger, the 6 over x will become closer and closer to zero.
02:25
And since x is positive, 6 over x is positive.
02:28
So our function will have a value greater than 2, but approaching 2.
02:34
So the limit as x approaches infinity of our function is equal to two, and it's doing that from above.
02:45
So if we want to graph it, we'll keep that in mind from above.
02:53
So that means we have a horizontal asymptote.
02:58
A horizontal asymptote, y equals two.
03:03
We can go ahead and draw that on right now.
03:07
So if this is one, this is two, then our asymptote would be right about here.
03:20
There's a horizontal asymptote.
03:23
And we know on the positive side, we are going to approach this asymptote from above.
03:33
How about on the negative side? as x approaches negative infinity, 6 over x is still going to approach 0.
03:41
We're going to have the larger x gets, even if it's in the negative direction, 6 over x will approach 0.
03:48
So our limit will still be 2.
03:50
So the limit as x approaches negative x of f of x is going to equal to as well...