WEBVTT
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The graph of a function π is shown.
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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.
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And d) π of minus two is equal to two.
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Letβs look at each statement in turn.
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Statement a) says that the limit of π of π₯ as π₯ tends to zero is equal to minus one.
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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.
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As π₯ tends to zero from the left, our function approaches minus one.
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We can therefore say that the left-hand limit of π of π₯ as π₯ tends to zero is minus one.
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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.
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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₯.
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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.
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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.
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Since these limits are not equal, then our statement a is false.
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Letβs look now at statement b.
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Statement b) says the limit as π₯ tends to zero from the positive direction of π of π₯ is equal to two.
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In fact, we saw already that as π₯ tends to zero from the positive direction, π of π₯ does indeed approach two.
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So statement b is actually correct.
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Letβs look now at statement c.
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Statement c) says the limit as π₯ tends to one of π of π₯ is equal to two.
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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.
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But this does not appear to be the case.
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As π₯ tends to one from the positive direction, the function does approach two.
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But as π₯ approaches one from the negative direction, π of π₯ does not approach two.
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So the right-hand and the left-hand limits as π₯ tends to one are not the same.
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Therefore, statement c is false.
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Now letβs look at the last statement, statement d.
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Statement d) simply says that π of minus two is equal to two.
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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.
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Hence, π of minus two does not equal two.
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So statement d is also false.
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The only true statement therefore is statement b.
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That the limit as π₯ tends to zero from the positive side of π of π₯ is equal to two.