Video Transcript
Find the limit as 𝑥 approaches two
of 𝑥 squared minus four all squared all divided by four 𝑥 minus eight.
In this question, we’re asked to
evaluate the limit as 𝑥 approaches two of a function. And if we were to distribute the
square over the parentheses in our numerator, we would see this is a rational
function. It’s the quotient of two
polynomials. And we know we can attempt to
evaluate this by direct substitution. And it’s always a good idea to use
direct substitution when it’s possible. So we’ll substitute 𝑥 is equal to
two in our function, giving us two squared minus four all squared all divided by
four times two minus eight.
But if we evaluate the expression
in our numerator and our denominator, we would get zero divided by zero. This is called an indeterminate
form. What this means is we can’t
determine our limit by using this method, so we’ll have to use some manipulation to
rewrite our limit in a form which we can evaluate the limit. And one way of doing this which
will work for rational functions is to use what we know about the factor
theorem. Substituting 𝑥 is equal to two
into the polynomial in our numerator gave us zero. Therefore, by the factor theorem,
𝑥 minus two is a factor of the polynomial in our numerator. The same is also true in our
denominator. So let’s try factoring the
numerator and the denominator of the function inside of our limit.
Factoring the denominator is
easy. We can take out the shared factor
of four. Doing this means we can rewrite our
limit as the limit as 𝑥 approaches two of 𝑥 squared minus four all squared all
divided by four times 𝑥 minus two. We now want to factor our
numerator. And to do this, we need to notice
inside of our parentheses, we have a difference between two squares. Recall, we know for a difference
between two squares, 𝑎 squared minus 𝑏 squared will be equal to 𝑎 minus 𝑏
multiplied by 𝑎 plus 𝑏. And we can see this is true in our
numerator. We’ll set 𝑎 equal to 𝑥 and 𝑏
equal to two. So by factoring our numerator using
difference between squares, we now have the limit as 𝑥 approaches two of 𝑥 minus
two times 𝑥 plus two all squared all divided by four times 𝑥 minus two.
However, we still can’t evaluate
this by using direct substitution. We would see we would get a factor
of zero in our numerator and a factor of zero in our denominator. So we’re still going to need to do
more manipulation. We’ll now distribute the square
over the parentheses in our numerator. Doing this, we get 𝑥 minus two all
squared multiplied by 𝑥 plus two all squared. And now we’re almost in a position
we can evaluate this by using direct substitution. What we want to do is cancel the
shared factor of 𝑥 minus two in our numerator and our denominator. If we did this, we would then get
an expression which doesn’t have a factor of 𝑥 minus two in our denominator, so we
could evaluate this by using direct substitution.
However, it’s worth reiterating why
we’re allowed to do this. When we take the limit as 𝑥
approaches two, we’re interested in what happens as 𝑥 gets closer and closer to
two. It doesn’t matter what happens when
𝑥 is equal to two. Therefore, we can assume 𝑥 minus
two is not equal to zero. And it won’t change the value of
our limit. Therefore, the limit given to us in
the question will be the same as the limit as 𝑥 approaches two of 𝑥 minus two
times 𝑥 plus two all squared all divided by four. And this is the limit as 𝑥
approaches two of a rational function. Or alternatively, it’s the limit as
𝑥 approaches two of a cubic polynomial, so we can do this by using direct
substitution.
So we substitute in 𝑥 is equal to
two. This gives us two minus two times
two plus two all squared all divided by four. And we can see in our numerator, we
have a factor of two minus two, which is equal to zero. However, our denominator is equal
to four. Therefore, this limit evaluates to
give us zero. And this is our final answer. We were able to show by using
algebraic manipulation and direct substitution the limit as 𝑥 approaches two of 𝑥
squared minus four all squared all divided by four 𝑥 minus eight is equal to
zero.
