Question Video: Understanding the Limit of a Function at a Point Mathematics • Second Year of Secondary School

Video Transcript

Which of the following statements is not the same as saying that the limit as 𝑥 approaches eight of 𝑓 of 𝑥 is equal to three? Is it option (A) we can make 𝑓 of 𝑥 as close as we like to three by taking 𝑥 sufficiently close to eight? Is it option (B) 𝑓 evaluated at three is equal to 𝑓 evaluated at eight? Option (C) as 𝑥 gets closer and closer to eight, 𝑓 of 𝑥 gets closer and closer to three. Or is it option (D) 𝑓 of 𝑥 approaches three as 𝑥 approaches eight?

In this question, we’re given four statements, and we need to determine which of these four statements is not the same as saying that the limit as 𝑥 approaches eight of 𝑓 of 𝑥 is equal to three. To answer this question, let’s start by recalling what we mean by the value of the limit of a function at a point. We recall that we say if the values of 𝑓 of 𝑥 approach some finite value of 𝐿, as the values of 𝑥 approach some value of 𝑎 from either side but not necessarily when 𝑥 is equal to 𝑎, then we say that the limit as 𝑥 approaches 𝑎 of 𝑓 of 𝑥 is equal to 𝐿. And we can directly use this definition to answer the question. However, let’s start by replacing the values of 𝑎 and 𝐿 in the definition with the values given in the question.

We’re taking the limit as 𝑥 approaches eight of our function 𝑓 of 𝑥, so we’ll set our value of 𝑎 equal to eight. And we say that this limit is equal to three, so we’ll set our value of 𝐿 equal to three. We can then update our definition. This now tells us since the limit as 𝑥 approaches eight of 𝑓 of 𝑥 is equal to three as the values of 𝑥 approach eight from either side but not necessarily when 𝑥 is equal to eight, then the values of 𝑓 of 𝑥 must be approaching three. And this is really useful because we can see that this statement is equivalent to three of our options.

First, we can make 𝑓 of 𝑥 as close as we like to three by taking 𝑥 sufficiently close to eight. This is a direct result from our definition. Our values of 𝑓 of 𝑥 are approaching three as our values of 𝑥 approach eight. And since 𝑓 of 𝑥 approaches three, we can make this as close as we like by taking 𝑥 sufficiently close to eight. So, option (A) is exactly the same as saying the limit as 𝑥 approaches eight of 𝑓 of 𝑥 is equal to three.

We can see the same is true in option (C). This is exactly the same as saying the limit as 𝑥 approaches eight of 𝑓 of 𝑥 is equal to three. It’s almost exactly the same as what is written. We only need to note one thing. When we say that our values of 𝑥 are getting closer and closer to eight, we do mean that 𝑥 can approach from either side and we don’t need to know what happens when 𝑥 is equal to eight. And this is exactly the same as our definition. As our values of 𝑥 approach eight from either side but not necessarily when 𝑥 is equal to eight, the values of 𝑓 of 𝑥 are approaching three.

Finally, in option (D), we see that 𝑓 of 𝑥 approaches three as 𝑥 approaches eight. And once again, this is exactly the same as what is written. So, this option is also the same as saying the limit as 𝑥 approaches eight of 𝑓 of 𝑥 is equal to three. Now, this is enough to answer our question. The other three options are all the same. So, option (B) must be different. However, for due diligence, let’s show this.

This statement says that the value of 𝑓 of three needs to be equal to 𝑓 evaluated at eight. Of course, we can immediately notice a few things wrong. For example, this statement does not tell us any information about the values of 𝑓 of 𝑥 as our values of 𝑥 get closer and closer to eight. And we can also notice in our definition we don’t need to know the value of our function when 𝑥 is equal to eight. The exact value of 𝑓 evaluated at eight will have no effect on the limit as 𝑥 approaches eight of 𝑓 of 𝑥. In particular, our function does not even need to be defined at 𝑥 is equal to eight.

So, knowing that the limit as 𝑥 approaches eight of 𝑓 of 𝑥 is equal to three won’t tell us any information about the function evaluated at eight or the function evaluated at three. So, we can say that option (B) saying that 𝑓 evaluated at three is equal to 𝑓 evaluated at eight is not the same as saying that the limit as 𝑥 approaches eight of 𝑓 of 𝑥 is equal to three.