### Video Transcript

Find the limit as π₯ approaches
negative two of π₯ to the eighth power minus 256 all divided by π₯ to the fourth
power minus 16.

We can see weβre asked to evaluate
the limit of the quotient of two polynomials. This is the limit of a rational
function. So we can try doing this by direct
substitution. We substitute π₯ is equal to
negative two into the function inside of our limit. This gives us negative two to the
eighth power minus 256 all divided by negative two to the fourth power minus 16. And if we evaluate this expression,
we get 256 minus 256 all divided by 16 minus 16, which simplifies to give us zero
over zero, which is an indeterminate form. This tells us we canβt evaluate our
limit by using direct substitution; weβll need to use a different method.

And since this is the limit of the
quotient of two polynomials, weβll try factoring our numerator and denominator. Letβs start with the polynomial in
our numerator. Thatβs π₯ to the eighth power minus
256. We can actually see this is the
difference between two squares. And it might be easier to see this
if we rewrite this as π₯ to the fourth power all squared minus 16 squared. Now we need to recall how we factor
a difference between squares. We know π squared minus π squared
is equal to π minus π multiplied by π plus π. So by setting our value of π to be
π₯ to the fourth power and π to be 16, we can factor the polynomial in our
numerator to be π₯ to the fourth power minus 16 multiplied by π₯ to the fourth power
plus 16.

And now we can see something
interesting. π₯ to the fourth power minus 16 is
a factor in our numerator, and itβs our denominator. In other words, weβve rewritten the
limit given to us in the question as the limit as π₯ approaches negative two of π₯
to the fourth power minus 16 multiplied by π₯ to the fourth power plus 16 all
divided by π₯ to the fourth power minus 16. Now, we want to cancel the shared
factor of π₯ to the fourth power minus 16 in both our numerator and our
denominator. And we can do this with
polynomials. Since weβre taking the limit as π₯
approaches negative two of this function, weβre only interested in what happens as
π₯ gets closer and closer to negative two, not what happens when π₯ is equal to
negative two. So we can cancel the shared factor
in our numerator and our denominator. It wonβt be equal to zero.

So by doing this, weβve shown our
limit is equal to the limit as π₯ approaches negative two of π₯ to the fourth power
plus 16. And this is the limit of a
polynomial, so we can evaluate this by using direct substitution. Substituting π₯ is equal to
negative two into our polynomial gives us negative two to the fourth power plus 16,
which we can calculate is equal to 32. Therefore, by using algebraic
manipulation and direct substitution, we were able to show the limit as π₯
approaches negative two of π₯ to the eighth power minus 256 all divided by π₯ to the
fourth power minus 16 is equal to 32.