### Video Transcript

The graph of a function π is shown. Which of the following statements about π is true? a) The limit of π of π₯ as π₯ tends to zero is negative one. b) The limit of π of π₯ as π₯ tends to zero from the positive direction is equal to two. c) The limit of π of π₯ as π₯ tends to one is equal to two. And d) π of minus two is equal to two.

Letβs look at each statement in turn. Statement a) says that the limit of π of π₯ as π₯ 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 π₯ tends to zero. As π₯ tends to zero from the left, our function approaches minus one. We can therefore say that the left-hand limit of π of π₯ as π₯ tends to zero is minus one. As π₯ tends to zero from the right-hand side, however, our function approaches two, so that the limit as π₯ tends zero from the positive direction is equal to two.

Now we know that, for a function π of π₯, the limit as π₯ tends to π of π of π₯ is equal to πΏ if and only if the left-hand limit as π₯ tends to π of π of π₯ is equal to πΏ is equal to the right-hand limit as π₯ tends to π of π of π₯. This means that πΏ is the limit as π₯ tends to π of π of π₯ only if the right- and left-hand limits are also equal to πΏ.

In statement a, as π₯ 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.

Letβs look now at statement b. Statement b) says the limit as π₯ tends to zero from the positive direction of π of π₯ is equal to two. In fact, we saw already that as π₯ tends to zero from the positive direction, π of π₯ does indeed approach two. So statement b is actually correct.

Letβs look now at statement c. Statement c) says the limit as π₯ tends to one of π of π₯ is equal to two. If the limit as π₯ tends to one of π of π₯ is equal to two, then the limit as π₯ tends to one from the positive direction must equal the limit as π₯ tends to one from the negative direction. But this does not appear to be the case. As π₯ tends to one from the positive direction, the function does approach two. But as π₯ approaches one from the negative direction, π of π₯ does not approach two. So the right-hand and the left-hand limits as π₯ tends to one are not the same. Therefore, statement c is false.

Now letβs look at the last statement, statement d. Statement d) simply says that π of minus two is equal to two. If we look at our graph, although the right- and left-hand limits as π₯ tends to two of π of π₯ are the same, the function itself does not exist at π₯ equals two. Hence, π 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 π₯ tends to zero from the positive side of π of π₯ is equal to two.