Video: Evaluating Limits from the Left and Right as x Approaches a Given Value

The graph of a function 𝑓 is shown. Which of the following statements about 𝑓 is true? [A] lim_(π‘₯ β†’ 0) 𝑓(π‘₯) = βˆ’1 [B] lim_(π‘₯ β†’ 0⁺) 𝑓(π‘₯) = 2 [C] lim_(π‘₯ β†’ 1) 𝑓(π‘₯) = 2 [D] 𝑓(βˆ’2) = 2

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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 π‘₯ as π‘₯ tends to zero is negative one. b) The limit of 𝑓 of π‘₯ as π‘₯ tends to zero from the positive direction is equal to two. c) The limit of 𝑓 of π‘₯ as π‘₯ 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 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 𝑓 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 𝑓 of π‘₯, the limit as π‘₯ tends to π‘Ž of 𝑓 of π‘₯ is equal to 𝐿 if and only if the left-hand limit as π‘₯ tends to π‘Ž of 𝑓 of π‘₯ is equal to 𝐿 is equal to the right-hand limit as π‘₯ tends to π‘Ž of 𝑓 of π‘₯. This means that 𝐿 is the limit as π‘₯ tends to π‘Ž of 𝑓 of π‘₯ only if the right- and left-hand limits are also equal to 𝐿.

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.

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

Let’s look now at statement c. Statement c) says the limit as π‘₯ tends to one of 𝑓 of π‘₯ is equal to two. If the limit as π‘₯ tends to one of 𝑓 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, 𝑓 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 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 π‘₯ tends to two of 𝑓 of π‘₯ are the same, the function itself does not exist at π‘₯ 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 π‘₯ tends to zero from the positive side of 𝑓 of π‘₯ is equal to two.

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