Question Video: Defining Limits of a Function Mathematics • Second Year of Secondary School

Video Transcript

True or false: If the limit as 𝑥 approaches one of 𝑓 of 𝑥 is equal to four, then it is possible that the function 𝑓 could be undefined at 𝑥 equals one.

We recall that this notation tells us that as 𝑥 gets closer and closer to one, as 𝑥 approaches one, the function itself gets closer and closer to four. So the function itself approaches four. And the reason that we use limits is because we sometimes have functions that are not defined at a given point.

For example, let’s take the function 𝑓 of 𝑥 equals 𝑒 to the power of negative one over 𝑥 squared. If we try to evaluate the function at 𝑥 equals zero, we get 𝑒 to the power of negative one over zero squared. But, of course, there’s a fraction here with the denominator of zero. So we would say that this is undefined.

Nevertheless, if we look closely at the graph of 𝑦 equals 𝑒 to the power of negative one over 𝑥 squared, we see that the function gets closer and closer to zero as 𝑥 itself gets closer and closer to zero. So, in fact, the limit as 𝑥 approaches zero of 𝑒 to the power of negative one over 𝑥 squared is zero. The function is undefined at 𝑥 equals zero, but it does have a limit.

And so, if we go back to our limit, we know that the limit of the function as 𝑥 gets closer and closer to one is four. But this doesn’t actually tell us anything about whether the function itself is defined at the point where 𝑥 equals one. And so, the statement is true. It is indeed possible that the function 𝑓 could be undefined at 𝑥 equals one. But we aren’t able to tell without any further information.