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Format: Layer, Start, End, Style, Name, MarginL, MarginR, MarginV, Effect, Text
Dialogue: 0,0:00:00.00,0:00:03.80,Default,,0000,0000,0000,,- [Instructor] In a previous\Nvideo, we explored the graphs
Dialogue: 0,0:00:00.00,0:00:00.88,Default,,0000,0000,0000,,of Y equals one over X\Nsquared and one over X.
Dialogue: 0,0:00:00.88,0:00:02.98,Default,,0000,0000,0000,,In a previous video we've\Nlooked at these graphs.
Dialogue: 0,0:00:02.98,0:00:05.35,Default,,0000,0000,0000,,This is Y is equal to one over X squared.
Dialogue: 0,0:00:05.35,0:00:07.84,Default,,0000,0000,0000,,This is Y is equal to one over X.
Dialogue: 0,0:00:07.84,0:00:09.70,Default,,0000,0000,0000,,And we explored what's the limit
Dialogue: 0,0:00:09.70,0:00:13.85,Default,,0000,0000,0000,,as X approaches zero in\Neither of those scenarios.
Dialogue: 0,0:00:13.85,0:00:15.82,Default,,0000,0000,0000,,And in this left scenario we saw
Dialogue: 0,0:00:15.82,0:00:18.34,Default,,0000,0000,0000,,as X becomes less and less negative,
Dialogue: 0,0:00:18.34,0:00:22.92,Default,,0000,0000,0000,,as it approaches zero\Nfrom the left hand side,
Dialogue: 0,0:00:22.92,0:00:26.18,Default,,0000,0000,0000,,the value of one over\NX squared is unbounded
Dialogue: 0,0:00:26.18,0:00:27.56,Default,,0000,0000,0000,,in the positive direction.
Dialogue: 0,0:00:27.56,0:00:30.93,Default,,0000,0000,0000,,And the same thing happens as\Nwe approach X from the right,
Dialogue: 0,0:00:30.93,0:00:32.53,Default,,0000,0000,0000,,as we become less and less positive
Dialogue: 0,0:00:32.53,0:00:34.17,Default,,0000,0000,0000,,but we are still positive,
Dialogue: 0,0:00:34.17,0:00:35.99,Default,,0000,0000,0000,,the value of one over X squared becomes
Dialogue: 0,0:00:35.99,0:00:38.01,Default,,0000,0000,0000,,unbounded in the positive direction.
Dialogue: 0,0:00:38.01,0:00:39.73,Default,,0000,0000,0000,,So in that video, we just said, "Hey,
Dialogue: 0,0:00:39.73,0:00:43.15,Default,,0000,0000,0000,,"one could say that this\Nlimit is unbounded."
Dialogue: 0,0:00:43.15,0:00:45.27,Default,,0000,0000,0000,,But what we're going\Nto do in this video is
Dialogue: 0,0:00:45.27,0:00:47.41,Default,,0000,0000,0000,,introduce new notation.
Dialogue: 0,0:00:47.41,0:00:49.15,Default,,0000,0000,0000,,Instead of just saying it's unbounded,
Dialogue: 0,0:00:49.15,0:00:51.16,Default,,0000,0000,0000,,we could say, "Hey, from\Nboth the left and the right
Dialogue: 0,0:00:51.16,0:00:53.59,Default,,0000,0000,0000,,it looks like we're going\Nto positive infinity".
Dialogue: 0,0:00:53.59,0:00:55.70,Default,,0000,0000,0000,,So we can introduce\Nthis notation of saying,
Dialogue: 0,0:00:55.70,0:00:58.32,Default,,0000,0000,0000,,"Hey, this is going to infinity",
Dialogue: 0,0:00:58.32,0:01:00.24,Default,,0000,0000,0000,,which you will sometimes see used.
Dialogue: 0,0:01:00.24,0:01:01.70,Default,,0000,0000,0000,,Some people would call this unbounded,
Dialogue: 0,0:01:01.70,0:01:03.16,Default,,0000,0000,0000,,some people say it does not exist
Dialogue: 0,0:01:03.16,0:01:05.73,Default,,0000,0000,0000,,because it's not approaching\Nsome finite value,
Dialogue: 0,0:01:05.73,0:01:07.63,Default,,0000,0000,0000,,while some people will use this notation
Dialogue: 0,0:01:07.63,0:01:10.22,Default,,0000,0000,0000,,of the limit going to infinity.
Dialogue: 0,0:01:10.22,0:01:11.76,Default,,0000,0000,0000,,But what about this scenario?
Dialogue: 0,0:01:11.76,0:01:14.22,Default,,0000,0000,0000,,Can we use our new notation here?
Dialogue: 0,0:01:14.22,0:01:18.31,Default,,0000,0000,0000,,Well, when we approach zero from the left,
Dialogue: 0,0:01:18.31,0:01:21.12,Default,,0000,0000,0000,,it looks like we're unbounded\Nin the negative direction,
Dialogue: 0,0:01:21.12,0:01:23.31,Default,,0000,0000,0000,,and when we approach zero from the right,
Dialogue: 0,0:01:23.31,0:01:26.26,Default,,0000,0000,0000,,we are unbounded in\Nthe positive direction.
Dialogue: 0,0:01:26.26,0:01:28.76,Default,,0000,0000,0000,,So, here you still could not say
Dialogue: 0,0:01:28.76,0:01:30.71,Default,,0000,0000,0000,,that the limit is approaching infinity
Dialogue: 0,0:01:30.71,0:01:32.44,Default,,0000,0000,0000,,because from the right\Nit's approaching infinity,
Dialogue: 0,0:01:32.44,0:01:34.66,Default,,0000,0000,0000,,but from the left it's\Napproaching negative infinity.
Dialogue: 0,0:01:34.66,0:01:39.66,Default,,0000,0000,0000,,So you would still say\Nthat this does not exist.
Dialogue: 0,0:01:39.76,0:01:42.34,Default,,0000,0000,0000,,You could do one sided limits here,
Dialogue: 0,0:01:42.34,0:01:43.67,Default,,0000,0000,0000,,which if you're not familiar with,
Dialogue: 0,0:01:43.67,0:01:45.78,Default,,0000,0000,0000,,I encourage you to review\Nit on Khan Academy.
Dialogue: 0,0:01:45.78,0:01:49.01,Default,,0000,0000,0000,,If you said the limit of one over X
Dialogue: 0,0:01:49.01,0:01:53.34,Default,,0000,0000,0000,,as X approaches zero\Nfrom the left hand side,
Dialogue: 0,0:01:53.34,0:01:55.81,Default,,0000,0000,0000,,from values less than zero,
Dialogue: 0,0:01:55.81,0:01:57.63,Default,,0000,0000,0000,,well then you would look at\Nthis right over here and say,
Dialogue: 0,0:01:57.63,0:01:59.54,Default,,0000,0000,0000,,"Well, look, it looks like we're going
Dialogue: 0,0:01:59.54,0:02:00.79,Default,,0000,0000,0000,,unbounded in the negative direction".
Dialogue: 0,0:02:00.79,0:02:04.27,Default,,0000,0000,0000,,So you would say this is\Nequal to negative infinity.
Dialogue: 0,0:02:04.27,0:02:09.27,Default,,0000,0000,0000,,And of course if you said the\Nlimit as X approaches zero
Dialogue: 0,0:02:09.67,0:02:12.70,Default,,0000,0000,0000,,from the right of one over X, well here
Dialogue: 0,0:02:12.70,0:02:14.50,Default,,0000,0000,0000,,you're unbounded in the positive direction
Dialogue: 0,0:02:14.50,0:02:17.65,Default,,0000,0000,0000,,so that's going to be\Nequal to positive infinity.
Dialogue: 0,0:02:17.65,0:02:19.76,Default,,0000,0000,0000,,Let's do an example\Nproblem from Khan Academy
Dialogue: 0,0:02:19.76,0:02:22.49,Default,,0000,0000,0000,,based on this idea and this notation.
Dialogue: 0,0:02:23.61,0:02:27.54,Default,,0000,0000,0000,,So here it says, consider\Ngraphs A, B, and C.
Dialogue: 0,0:02:27.54,0:02:30.47,Default,,0000,0000,0000,,The dashed lines represent asymptotes.
Dialogue: 0,0:02:30.47,0:02:33.26,Default,,0000,0000,0000,,Which of the graphs agree\Nwith this statement,
Dialogue: 0,0:02:33.26,0:02:36.16,Default,,0000,0000,0000,,that the limit as X approaches 1 of H of X
Dialogue: 0,0:02:36.16,0:02:37.48,Default,,0000,0000,0000,,is equal to infinity?
Dialogue: 0,0:02:37.48,0:02:39.98,Default,,0000,0000,0000,,Pause this video and see\Nif you can figure it out.
Dialogue: 0,0:02:40.94,0:02:42.35,Default,,0000,0000,0000,,Alright, let's go through each of these.
Dialogue: 0,0:02:42.35,0:02:44.85,Default,,0000,0000,0000,,So we want to think about\Nwhat happens at X equals one.
Dialogue: 0,0:02:44.85,0:02:47.86,Default,,0000,0000,0000,,So that's right over here on graph A.
Dialogue: 0,0:02:47.86,0:02:49.88,Default,,0000,0000,0000,,So as we approach X equals one,
Dialogue: 0,0:02:49.88,0:02:52.12,Default,,0000,0000,0000,,so let me write this, so the limit,
Dialogue: 0,0:02:52.12,0:02:53.86,Default,,0000,0000,0000,,let me do this for the different graphs.
Dialogue: 0,0:02:53.86,0:02:58.75,Default,,0000,0000,0000,,So, for graph A, the\Nlimit as x approaches one
Dialogue: 0,0:02:59.68,0:03:02.36,Default,,0000,0000,0000,,from the left, that looks like
Dialogue: 0,0:03:02.36,0:03:04.16,Default,,0000,0000,0000,,it's unbounded in the positive direction.
Dialogue: 0,0:03:04.16,0:03:07.09,Default,,0000,0000,0000,,That equals infinity and the limit
Dialogue: 0,0:03:07.09,0:03:11.53,Default,,0000,0000,0000,,as X approaches one from the right,
Dialogue: 0,0:03:11.53,0:03:14.02,Default,,0000,0000,0000,,well that looks like it's\Ngoing to negative infinity.
Dialogue: 0,0:03:14.02,0:03:15.97,Default,,0000,0000,0000,,That equals negative infinity.
Dialogue: 0,0:03:15.97,0:03:18.77,Default,,0000,0000,0000,,And since these are going\Nin two different directions,
Dialogue: 0,0:03:18.77,0:03:19.86,Default,,0000,0000,0000,,you wouldn't be able to say that
Dialogue: 0,0:03:19.86,0:03:21.42,Default,,0000,0000,0000,,the limit as X approaches one
Dialogue: 0,0:03:21.42,0:03:23.45,Default,,0000,0000,0000,,from both directions is equal to infinity.
Dialogue: 0,0:03:23.45,0:03:25.70,Default,,0000,0000,0000,,So I would rule this one out.
Dialogue: 0,0:03:25.70,0:03:27.71,Default,,0000,0000,0000,,Now let's look at choice B.
Dialogue: 0,0:03:27.71,0:03:32.71,Default,,0000,0000,0000,,What's the limit as X\Napproaches one from the left?
Dialogue: 0,0:03:33.22,0:03:36.25,Default,,0000,0000,0000,,And of course these are of H of X.
Dialogue: 0,0:03:36.25,0:03:37.61,Default,,0000,0000,0000,,Gotta write that down.
Dialogue: 0,0:03:37.61,0:03:40.97,Default,,0000,0000,0000,,So, of H of X right over here.
Dialogue: 0,0:03:40.97,0:03:43.59,Default,,0000,0000,0000,,Well, as we approach from the left,
Dialogue: 0,0:03:43.59,0:03:47.39,Default,,0000,0000,0000,,looks like we're going\Nto positive infinity.
Dialogue: 0,0:03:47.39,0:03:50.74,Default,,0000,0000,0000,,And it looks like the limit of H of X
Dialogue: 0,0:03:50.74,0:03:54.22,Default,,0000,0000,0000,,as we approach one from the right is
Dialogue: 0,0:03:54.22,0:03:56.86,Default,,0000,0000,0000,,also going to positive infinity.
Dialogue: 0,0:03:56.86,0:03:58.71,Default,,0000,0000,0000,,And so, since we're\Napproaching you could say
Dialogue: 0,0:03:58.71,0:04:02.63,Default,,0000,0000,0000,,the same direction of infinity,\Nyou could say this for B.
Dialogue: 0,0:04:02.63,0:04:04.49,Default,,0000,0000,0000,,So B meets the constraints, but
Dialogue: 0,0:04:04.49,0:04:06.73,Default,,0000,0000,0000,,let's just check C to make sure.
Dialogue: 0,0:04:06.73,0:04:09.89,Default,,0000,0000,0000,,Well, you can see very\Nclearly X equals one,
Dialogue: 0,0:04:09.89,0:04:11.23,Default,,0000,0000,0000,,that as we approach it from the left,
Dialogue: 0,0:04:11.23,0:04:12.49,Default,,0000,0000,0000,,we go to negative infinity,
Dialogue: 0,0:04:12.49,0:04:14.71,Default,,0000,0000,0000,,and as we approach from the right,
Dialogue: 0,0:04:14.71,0:04:15.54,Default,,0000,0000,0000,,we got to positive infinity.
Dialogue: 0,0:04:16.40,0:04:18.74,Default,,0000,0000,0000,,So this, once again,\Nwould not be approaching
Dialogue: 0,0:04:18.74,0:04:19.88,Default,,0000,0000,0000,,the same infinity.
Dialogue: 0,0:04:19.88,0:04:22.29,Default,,0000,0000,0000,,So you would rule this one out, as well.