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AP Calc: LIM‑4 (EU), LIM‑4.A (LO), LIM‑4.A.1 (EK), LIM‑4.A.2 (EK)

We need to evaluate the limit,
as x approaches infinity, of 4x squared minus 5x, all of that
over 1 minus 3x squared. So infinity is kind
of a strange number. You can't just plug in infinity
and see what happens. But if you wanted to evaluate
this limit, what you might try to do is just evaluate-- if you
want to find the limit as this numerator approaches infinity,
you put in really large numbers there, and you're going to see
that it approaches infinity. That the numerator
approaches infinity as x approaches infinity. And if you put really large
numbers in the denominator, you're going to see
that that also-- well, not quite infinity. 3x squared will approach
infinity, but we're subtracting it. If you subtract infinity from
some non-infinite number, it's going to be negative infinity. So if you were to just kind
of evaluate it at infinity, the numerator, you would
get positive infinity. The denominator, you would
get negative infinity. So I'll write it like this. Negative infinity. And that's one of the
indeterminate forms that L'Hopital's Rule
can be applied to. And you're probably saying,
hey, Sal, why are we even using L'Hopital's Rule? I know how to do this
without L'Hopital's Rule. And you probably
do, or you should. And we'll do that in a second. But I just wanted to show you
that L'Hopital's Rule also works for this type of problem,
and I really just wanted to show you an example that had a
infinity over negative or positive infinity
indeterminate form. But let's apply
L'Hopital's Rule here. So if this limit exists, or if
the limit of their derivatives exist, then this limit's going
to be equal to the limit as x approaches infinity of the
derivative of the numerator. So the derivative of the
numerator is-- the derivative of 4x squared is 8x minus 5
over-- the derivative of the denominator is, well,
derivative of 1 is 0. Derivative of negative 3x
squared is negative 6x. And once again, when you
evaluated infinity, the numerator is going to
approach infinity. And the denominator is
approaching negative infinity. Negative 6 times infinity
is negative infinity. So this is negative infinity. So let's apply
L'Hopital's Rule again. So if the limit of these guys'
derivatives exist-- or the rational function of the
derivative of this guy divided by the derivative of that guy--
if that exists, then this limit's going to be equal to
the limit as x approaches infinity of-- arbitrarily
switch colors-- derivative of 8x minus 5 is just 8. Derivative of negative
6x is negative 6. And this is just going to be--
this is just a constant here. So it doesn't matter what limit
you're approaching, this is just going to equal this value. Which is what? If we put it in lowest
common form, or simplified form, it's negative 4/3. So this limit exists. This was an indeterminate form. And the limit of this
function's derivative over this function's derivative exists,
so this limit must also equal negative 4/3. And by that same argument,
that limit also must be equal to negative 4/3. And for those of you who
say, hey, we already knew how to do this. We could just factor
out an x squared. You are absolutely right. And I'll show you
that right here. Just to show you that it's
not the only-- you know, L'Hopital's Rule is not
the only game in town. And frankly, for this type of
problem, my first reaction probably wouldn't have been to
use L'Hopital's Rule first. You could have said that that
first limit-- so the limit as x approaches infinity of 4x
squared minus 5x over 1 minus 3x squared is equal to the
limit as x approaches infinity. Let me draw a little line here,
to show you that this is equal to that, not to this
thing over here. This is equal to the limit
as x approaches infinity. Let's factor out an x squared
out of the numerator and the denominator. So you have an x squared
times 4 minus 5 over x. Right? x squared times 5
over x is going to be 5x. Divided by-- let's factor out
an x out of the numerator. So x squared times 1
over x squared minus 3. And then these x
squareds cancel out. So this is going to be equal
to the limit as x approaches infinity of 4 minus 5 over x
over 1 over x squared minus 3. And what's this going
to be equal to? Well, as x approaches
infinity-- 5 divided by infinity-- this term
is going to be 0. Super duper infinitely
large denominator, this is going to be 0. That is going to approach 0. And same argument. This right here is
going to approach 0. All you're left with is
a 4 and a negative 3. So this is going to be
equal to negative 4 over a negative 3, or negative 4/3. So you didn't have to do
use L'Hopital's Rule for this problem.

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