Video Transcript
Find the limit as 𝑥 approaches two
of eight 𝑥 cubed minus 64 divided by 𝑥 squared minus four.
Here, we have a function, which
we’ll call 𝑓 of 𝑥. Given that this is a rational
function, the first thing we may try is a direct substitution of 𝑥 equals two into
our function. Here, we have performed the
substitution. And when we evaluate our answer, we
find that we’re left with the indeterminate form of zero over zero. Instead, we’re gonna need to move
on to a different method based on factorisation.
For this method, we first note that
our function 𝑓 of 𝑥 is in the form 𝑃 of 𝑥 over 𝑄 of 𝑥, where both 𝑃 and 𝑄
are polynomial functions. By inspecting the numerator of our
quotient, we noticed that both the terms have a factor of eight. We can, therefore, factorise our
numerator as eight times 𝑥 cubed minus eight. And given that this eight is a
constant, we can take it outside of our limit as follows. If instead we find the limit as 𝑥
approaches two of 𝑥 cubed minus eight divided by 𝑥 squared minus four and multiply
this entire thing by our constant eight, this will give us the same answer.
To proceed, we can then notice that
the eight in our numerator and the four in our denominator can both be expressed as
powers of two, which are two cubed and two squared, respectively. After doing this, we see that our
limit now takes the following form. Here, we’ll say that the top half
of our quotient is equal to the difference of two 𝑛th powers, with 𝑛 being three,
and the bottom half of our quotient is equal to the difference of two 𝑚th powers,
with 𝑚 being two.
Given this form, we’re able to use
the following general rule, which tells us that the limit will be equal to 𝑛 over
𝑚 times 𝑎 to the power of 𝑛 minus 𝑚. At this point, you may notice that
both the top and bottom half of our quotient would have a common factor of 𝑥 minus
two. We could, instead, cancel this
common factor and proceeded by refactorising. However, this general rule allows
us to move directly to our limit.
In cases, where 𝑛 or 𝑚 are large,
this helps us potentially avoid a lengthy or time-consuming refactorisation. Inputting our values into our
general rule, where 𝑎 is two, 𝑛 is three, and 𝑚 is also two, we find that our
limit is three over two multiplied by two to the power of three minus two. We also mustn’t forget to multiply
this entire thing by the eight which we took out of our limit.
Three minus two is, of course, just
one. And so, we can simplify this by
cancelling the two and the one over two. We then find that our answer is
equal to eight times three, which 24. We have now found that the limit,
as 𝑥 approaches two of our function 𝑓 of 𝑥, is equal to 24. And we have answered our
question.
