Video Transcript
Discuss the existence of the limit
of 𝑓 of 𝑥 as 𝑥 approaches negative one given that 𝑓 of 𝑥 is equal to 𝑥 minus
four if 𝑥 is less than negative one, and 𝑓 of 𝑥 is equal to 20 if 𝑥 is greater
than negative one.
The question is asking us to
discuss the existence of the limit of 𝑓 of 𝑥 as 𝑥 is approaching negative
one. And we know that for any function
𝑔 of 𝑥, the limit of 𝑔 of 𝑥 as 𝑥 approaches 𝑎 exists if both the left-hand
limit and the right-hand limit exist and are equal. In our case, we are testing the
function 𝑓 of 𝑥 as 𝑥 is approaching negative one. So we can add this information into
our definition here. This means that there are three
parts to checking the existence of the limit in our question.
First, we need to check that the
limit of 𝑓 of 𝑥 as 𝑥 approaches negative one from the right exists. Second, we need to check that the
limit of 𝑓 of 𝑥 as 𝑥 approaches negative one from the left exists. Third, we need to check that if
both of these limits exist, then they’re equal. So we’re going to need to check the
limit from the left and the limit from the right. We will start with the limit as 𝑥
approaches negative one from the right.
Since we have that 𝑥 is
approaching negative one from the right, we must have that 𝑥 is greater than
negative one. We also know that if two functions
𝑓 of 𝑥 and 𝑔 of 𝑥 are equal for 𝑥 is greater than 𝑎, then their right limits
as 𝑥 approaches 𝑎 must be equal. From the question, we have that the
function 𝑓 of 𝑥 is equal to 20 when 𝑥 is greater than negative one. Therefore, since 𝑓 of 𝑥 is equal
to 20 when 𝑥 is greater than negative one, we can replace the 𝑓 of 𝑥 in our
right-hand limit with 20. We also know that for any constant
𝑘, the limit as 𝑥 approaches 𝑎 of the constant 𝑘 is just equal to 𝑘. Therefore, we can evaluate the
limit as 𝑥 approaches negative one of 20 to just be 20.
We can now use a similar method to
calculate the left-hand limit. We will start by restating our
rule. If two functions 𝑓 of 𝑥 and 𝑔 of
𝑥 agree everywhere, where 𝑥 is less than 𝑎, then the limit of 𝑓 of 𝑥 as 𝑥
approaches 𝑎 from the left must be equal to the limit of 𝑔 of 𝑥 as 𝑥 approaches
𝑎 from the left. Since 𝑥 is approaching from the
left, we must have that 𝑥 is less than negative one. And from the question, we can see
that when 𝑥 is less than negative one, we must have that 𝑓 of 𝑥 is equal to 𝑥
minus four. So using our rule, we can rewrite
our left limit as the limit of 𝑥 minus four as 𝑥 approaches negative one from the
left.
We know that we can evaluate the
limit of a polynomial function 𝑓 by using direct substitution. In our case, we want to find the
limit of 𝑥 minus four which we know is a polynomial. So we can use direct substitution
to find the limit of 𝑥 minus four as 𝑥 approaches negative one. So we substitute our value of
negative one into the function to get negative one minus four. Which we can then evaluate as
negative five. Since the left-hand limit must be
equal to the limit, we must have that the limit of 𝑥 minus four as 𝑥 approaches
negative one is equal to negative five. What is interesting here is if we
sketch a graph of our function 𝑓 of 𝑥. So let’s clear some space.
We have when 𝑥 is greater than
negative one, we have that 𝑓 of 𝑥 is equal to 20. So this gives us the line 𝑦 equals
20 for 𝑥 is greater than negative one. And when 𝑥 is less than negative
one, we have that 𝑓 of 𝑥 is equal to 𝑥 minus four. So we sketch the line 𝑦 equals 𝑥
minus four. But we restrict the domain to 𝑥
being less than negative one. So we can see that our function 𝑓
of 𝑥 has no output when 𝑥 is equal to negative one. And we can also see the two limits
that we have calculated on our graph. We can see that as 𝑥 approaches
negative one from the left, we get closer and closer to our output value of negative
five. And we can see as 𝑥 approaches
negative one from the right, we stay at a constant value of 20.
From our calculations, we can see
that the left-hand limit and the right-hand limit of 𝑓 of 𝑥 as 𝑥 approaches
negative one are not equal. And we can see graphically that the
left-hand limit and the right-hand limit of 𝑓 of 𝑥 as 𝑥 approaches negative one
are not equal. Therefore, we can conclude that the
limit of 𝑓 of 𝑥 as 𝑥 tends to negative one does not exist. Because the left-hand limit and the
right-hand limit of 𝑓 of 𝑥 as 𝑥 approaches negative one are not equal.
