Video Transcript
Determine the limit as 𝑥 approaches negative four from the left of 𝑓 of 𝑥 if it exists.
We’re given a sketch of the function 𝑓 of 𝑥. We need to use this sketch to determine whether the limit as 𝑥 approaches negative four from the left of 𝑓 of 𝑥 exists. And if it does exist, we need to determine the value of this limit. Let’s start by recalling what we mean by the limit as 𝑥 approaches negative four from the left of a function 𝑓 of 𝑥. This is the value 𝑓 of 𝑥 approaches as 𝑥 tends to negative four and 𝑥 is less than negative four. And it’s worth pointing out that sometimes our outputs 𝑓 of 𝑥 don’t approach any one value.
For example, our outputs 𝑓 of 𝑥 could keep getting larger and larger going above any value. They could also decrease more and more until they go below any value. And they could also oscillate around several values, never getting closer to one specific value. In all of these cases, we don’t define the limit in this way. We just say that this limit does not exist. However, to check whether the limit does exist, we need to consider this definition. We want to see what happens as our input values of 𝑥 get closer and closer to negative four from the left. So this means all of our inputs will be less than negative four and we want to see what happens to our output values of 𝑓 of 𝑥 when this happens.
Remember, our inputs will be on the 𝑥-axis. Since we want to know what happens to our function 𝑓 of 𝑥 around our inputs of negative four, let’s mark 𝑥 is equal to negative four on our diagram. Remember, we want to know what happens to our outputs as 𝑥 approaches negative four from the left. So our input values of 𝑥 will all be less than negative four. And immediately, we can see a couple of interesting things. First, our function is not defined when 𝑥 is less than negative six. So we’ll need to pick values of 𝑥 greater than negative six and values of 𝑥 less than negative four.
Let’s start with 𝑥 is equal to negative six. We can see from our diagram that 𝑓 of negative six is equal to eight. We can do the same when 𝑥 is equal to negative five. We get 𝑓 evaluated at negative five is equal to three. We want to use more and more points which are getting closer and closer to 𝑥 is equal to negative four. And as we take more and more points getting closer and closer to negative four, we can see that our outputs are getting closer and closer to negative two.
And there’s one more thing we might be worried about. When 𝑥 is equal to negative four, our function 𝑓 of 𝑥 is not equal to negative two. In fact, it’s equal to negative four. But in actual fact, the value of 𝑓 evaluated at negative four does not matter. This is because we said we want to know what happens as 𝑥 tends to negative four. This means 𝑥 is getting closer and closer to negative four. But our input values of 𝑥 are never equal to negative four.
Therefore, because we showed as our values of 𝑥 get closer and closer to negative four from the left, our outputs 𝑓 of 𝑥 get closer and closer to negative two, we can conclude the limit as 𝑥 approaches negative four from the left of 𝑓 of 𝑥 is equal to negative two.
