Video Transcript
Given that the limit as 𝑥 approaches one of 𝑓 of 𝑥 is equal to four, which of the following statements must be true? Is it option (A) 𝑓 evaluated at one is equal to four? Option (B) 𝑓 evaluated at four is equal to one. Option (C) 𝑓 evaluated at one is not equal to four. Option (D) 𝑓 evaluated at four is not equal to one. Or is it option (E) none of the above?
In this question, we’re given a function 𝑓 of 𝑥, and we’re told that the limit of our function 𝑓 of 𝑥 as 𝑥 approaches one is equal to four. We need to use this information to determine which of five given statements is true. To do this, let’s start by recalling what we mean by the limit of a function at a point. We recall that we say if the values of our function 𝑓 of 𝑥 approach some finite value of 𝐿 as the values of 𝑥 approach 𝑎 from both sides, 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.
In the question, we’re told the limit as 𝑥 approaches one of 𝑓 of 𝑥 is equal to four. So let’s set the value of 𝑎 equal to one in this definition and 𝐿 equal to four. This gives us the following definition. The values of 𝑓 of 𝑥 are approaching four as the values of 𝑥 approach one from either side, but not necessarily when 𝑥 is equal to one. And we can notice something interesting. We’re not told any specific output value of our function at an input value. We’re only given certain pieces of information about the outputs. For example, the outputs are approaching four as our values of 𝑥 approach one from either side. But we don’t know what these outputs are. We just know they get closer and closer to four.
Similarly, we’re specifically told in the definition we’re not interested in the output values of our function when 𝑥 is equal to one. And we can use this to conclude that options (A) and (C) cannot be correct. We can equally use this to prove that options (B) and (D) can also not be correct. One way of doing this is to note the only piece of information we’re given about our function is what happens to the output values of our function as our values of 𝑥 get closer and closer to one. So we’re only told information about the output values of our function around 𝑥 is equal to one. This allows us to conclude the output values of our function can be any value they want at four. For example, they could be equal to one, two, or our function might not even be defined.
Hence, we were able to show the answer must be option (E): none of the above statements must be true.
