Video Transcript
Find the limit of nine minus
eight 𝑥 plus six 𝑥 squared minus two 𝑥 cubed as 𝑥 approaches negative
infinity.
The first thing we might be
tempted to do is to write this limit of a sum as the sum of some limits. We can then evaluate each of
these limits one-by-one. The limit of the constant
function nine is just nine. What can we say about the limit
of eight 𝑥 as 𝑥 approaches negative infinity? Well, with the graph of 𝑦
equals eight 𝑥 in mind, we can see that as 𝑥 decreases without bound, 𝑦 also
decreases without bound. And so, the limit of eight 𝑥,
as 𝑥 approaches negative infinity, is negative infinity.
How about the limit of six 𝑥
squared as 𝑥 approaches negative infinity? Again, we have the graph in
mind, and we see that as 𝑥 decreases without bound, 𝑦 increases without
bound. So, this limit is infinity. And finally, the limit of two
𝑥 cubed as 𝑥 approaches negative infinity, we know what a cubic curve looks
like. And we see that as 𝑥
approaches negative infinity, 𝑦 also approaches negative infinity. This limit is negative
infinity.
So, it looks like our limit is
nine minus negative infinity plus infinity minus negative infinity. And if we treat infinity like a
number, we can write minus negative infinity as plus infinity, getting nine plus
infinity plus infinity plus infinity. And this sum is equal to
infinity. Now, we have to be slightly
careful about manipulating infinity in this way. But it turns out that all of
these steps are okay in this situation. We got lucky though that we
didn’t have any minus signs left at the end, as infinity minus infinity is
undefined.
For various reasons then, it
might be worth seeing how to solve this problem in a different way. What we do instead is we factor
out the highest power of 𝑥, that’s 𝑥 cubed, from inside the limit. This gives us the limit of 𝑥
cubed times nine 𝑥 to the negative three minus eight 𝑥 to the negative two
plus six 𝑥 to the negative one minus two as 𝑥 approaches negative
infinity. The limit of a product is the
product of the limits.
Now, what’s the limit of 𝑥
cubed as 𝑥 approaches negative infinity? Well, we can slightly modify
our graph and call this the graph of 𝑦 equals 𝑥 cubed instead. And we’ll see that this limit
is negative infinity. How about this limit? Well, these terms with negative
powers of 𝑥 contribute nothing, and so we’re left with just the limit of
negative two as 𝑥 approaches negative infinity. And this is, of course, just
negative two. The only bit of infinity
manipulation we need to do is to multiply negative infinity by negative two. The minus signs cancel. And we get just infinity.
Alternatively, we could’ve
factored the whole term negative two 𝑥 cubed out of the limit. And then, the value of the
second limit in our product would just be one. We can easily show that the
first limit in the product is infinity. And you might be more willing
to believe that infinity times one is infinity than you were to believe that
negative infinity times negative two is infinity.
Using this method, we can show
that the limit of a polynomial, as 𝑥 approaches positive or negative infinity,
is just the limit of the highest-degree term of that polynomial as 𝑥 approaches
positive or negative infinity. Then, all we have to do is look
at, or imagine, a graph of this monomial function.
