Video Transcript
The graph of a function 𝑓 is shown. Which of the following statements about 𝑓 is true? a) The limit of 𝑓 of 𝑥 as 𝑥 tends to zero is negative one. b) The limit of 𝑓 of 𝑥 as 𝑥 tends to zero from the positive direction is equal to two. c) The limit of 𝑓 of 𝑥 as 𝑥 tends to one is equal to two. And d) 𝑓 of minus two is equal to two.
Let’s look at each statement in turn. Statement a) says that the limit of 𝑓 of 𝑥 as 𝑥 tends to zero is equal to minus one. To determine whether this is true or not, we need to look at both the left-hand and the right-hand limits as 𝑥 tends to zero. As 𝑥 tends to zero from the left, our function approaches minus one. We can therefore say that the left-hand limit of 𝑓 of 𝑥 as 𝑥 tends to zero is minus one. As 𝑥 tends to zero from the right-hand side, however, our function approaches two, so that the limit as 𝑥 tends zero from the positive direction is equal to two.
Now we know that, for a function 𝑓 of 𝑥, the limit as 𝑥 tends to 𝑎 of 𝑓 of 𝑥 is equal to 𝐿 if and only if the left-hand limit as 𝑥 tends to 𝑎 of 𝑓 of 𝑥 is equal to 𝐿 is equal to the right-hand limit as 𝑥 tends to 𝑎 of 𝑓 of 𝑥. This means that 𝐿 is the limit as 𝑥 tends to 𝑎 of 𝑓 of 𝑥 only if the right- and left-hand limits are also equal to 𝐿.
In statement a, as 𝑥 tends to zero, the left-hand limit is equal to negative one and the right-hand limit is equal to two. Since these limits are not equal, then our statement a is false.
Let’s look now at statement b. Statement b) says the limit as 𝑥 tends to zero from the positive direction of 𝑓 of 𝑥 is equal to two. In fact, we saw already that as 𝑥 tends to zero from the positive direction, 𝑓 of 𝑥 does indeed approach two. So statement b is actually correct.
Let’s look now at statement c. Statement c) says the limit as 𝑥 tends to one of 𝑓 of 𝑥 is equal to two. If the limit as 𝑥 tends to one of 𝑓 of 𝑥 is equal to two, then the limit as 𝑥 tends to one from the positive direction must equal the limit as 𝑥 tends to one from the negative direction. But this does not appear to be the case. As 𝑥 tends to one from the positive direction, the function does approach two. But as 𝑥 approaches one from the negative direction, 𝑓 of 𝑥 does not approach two. So the right-hand and the left-hand limits as 𝑥 tends to one are not the same. Therefore, statement c is false.
Now let’s look at the last statement, statement d. Statement d) simply says that 𝑓 of minus two is equal to two. If we look at our graph, although the right- and left-hand limits as 𝑥 tends to two of 𝑓 of 𝑥 are the same, the function itself does not exist at 𝑥 equals two. Hence, 𝑓 of minus two does not equal two. So statement d is also false.
The only true statement therefore is statement b. That the limit as 𝑥 tends to zero from the positive side of 𝑓 of 𝑥 is equal to two.
