Video Transcript
Discuss the existence of the limit
as 𝑥 approaches two of one over the absolute value of 𝑥 minus two.
In this question, we are given a
limit and asked to consider the existence of this limit. We can recall that for a limit to
exist, we need the outputs of the function to approach a finite value as the inputs
approach the point from either side. This is equivalent to saying that
we need the left and right limit of the function at this point to both exist and
both be equal.
There are many ways in which we can
analyze this limit. For instance, we can sketch the
graph of this function by first noting it can be rewritten as the absolute value of
one over 𝑥 minus two. We can then sketch a graph of this
function by noting that it is a transformation of the graph of the reciprocal
function. We translate the graph two units to
the right and then reflect any part of the curve below the 𝑥-axis through the
𝑥-axis.
Applying these transformations to
the graph of the reciprocal curve gives us the following sketch. We can note that the vertical
asymptote is translated two units to the right to the line 𝑥 equals two and the
horizontal asymptote is still at 𝑦 equals zero. We can use this sketch to analyze
the left and right limits as 𝑥 approaches two for this function. First, we can note that as the
values of 𝑥 approach two from the left, the curve is unbounded and approaches
positive ∞. We can therefore say that the limit
as 𝑥 approaches two from the left of one over the absolute value of 𝑥 minus two is
∞. It is also worth reiterating here
that saying a limit is equal to ∞ also means that it does not exist. However, it is useful to note that
the limit is ∞.
We have a very similar story if we
analyze the graph with values of 𝑥 approaching two from the right. We see that the graph is unbounded
towards positive ∞, so the outputs of the function approach positive ∞ when 𝑥
approaches two from the right. So, once again, we can say that the
limit as 𝑥 approaches two from the right of this function does not exist. However, we can say that its limit
is positive ∞.
Since the left and right limit are
both ∞, we can say that the limit is equal to ∞. However, this does mean that the
limit does not exist. Hence, the limit does not exist,
but the limit as 𝑥 approaches two of one over the absolute value of 𝑥 minus two is
equal to ∞.
