[Script Info]
Title: 
[Events]
Format: Layer, Start, End, Style, Name, MarginL, MarginR, MarginV, Effect, Text
Dialogue: 0,0:00:01.43,0:00:05.26,Default,,0000,0000,0000,,In this video, we'll define what\Nit means for a function F to
Dialogue: 0,0:00:05.26,0:00:07.62,Default,,0000,0000,0000,,tend to a limit as X tends to
Dialogue: 0,0:00:07.62,0:00:12.08,Default,,0000,0000,0000,,Infinity. We also define what it\Nmeans for its tend to limit as X
Dialogue: 0,0:00:12.08,0:00:15.02,Default,,0000,0000,0000,,tends to minus Infinity, and as\NX tends to a real number.
Dialogue: 0,0:00:17.44,0:00:21.06,Default,,0000,0000,0000,,Remember from the video limits\Nof sequences, we define what it
Dialogue: 0,0:00:21.06,0:00:25.99,Default,,0000,0000,0000,,meant for a sequence to tend to\NA to a limit as N tends to
Dialogue: 0,0:00:25.99,0:00:32.27,Default,,0000,0000,0000,,Infinity, we said. Sequence\Nwired tends to limit L
Dialogue: 0,0:00:32.27,0:00:36.17,Default,,0000,0000,0000,,as an tends to Infinity.
Dialogue: 0,0:00:37.19,0:00:40.11,Default,,0000,0000,0000,,If however, small a number I
Dialogue: 0,0:00:40.11,0:00:46.50,Default,,0000,0000,0000,,chose. The sequence YN would get\Nthat close to L and stay that
Dialogue: 0,0:00:46.50,0:00:51.96,Default,,0000,0000,0000,,close. One of our examples\Nwas the sequence YN equals
Dialogue: 0,0:00:51.96,0:00:53.35,Default,,0000,0000,0000,,one over N.
Dialogue: 0,0:00:54.48,0:00:56.92,Default,,0000,0000,0000,,I just put a graph of this.
Dialogue: 0,0:01:01.72,0:01:05.44,Default,,0000,0000,0000,,So her plot the points\Nof why and which was
Dialogue: 0,0:01:05.44,0:01:07.30,Default,,0000,0000,0000,,one equals one over N.
Dialogue: 0,0:01:08.40,0:01:11.92,Default,,0000,0000,0000,,It looks like this sort of it 1.
Dialogue: 0,0:01:12.50,0:01:16.24,Default,,0000,0000,0000,,And then two was a half. Then it
Dialogue: 0,0:01:16.24,0:01:23.16,Default,,0000,0000,0000,,went 1/3. Now this\Nsequence tended to
Dialogue: 0,0:01:23.16,0:01:26.38,Default,,0000,0000,0000,,0 because eventually.
Dialogue: 0,0:01:26.93,0:01:29.39,Default,,0000,0000,0000,,The points in the sequence get
Dialogue: 0,0:01:29.39,0:01:34.84,Default,,0000,0000,0000,,as close. Zeros alike, so have a\Nsmaller distance I choose.
Dialogue: 0,0:01:38.01,0:01:41.48,Default,,0000,0000,0000,,The points in the sequence\Nwill eventually get within
Dialogue: 0,0:01:41.48,0:01:44.56,Default,,0000,0000,0000,,that boundary from zero and\Nstay in there.
Dialogue: 0,0:01:45.71,0:01:48.90,Default,,0000,0000,0000,,The functions are defined\Nin a similar way to limits
Dialogue: 0,0:01:48.90,0:01:49.54,Default,,0000,0000,0000,,of sequences.
Dialogue: 0,0:01:50.95,0:01:54.18,Default,,0000,0000,0000,,If we have a function like FX\Nequals one over X.
Dialogue: 0,0:01:54.74,0:01:57.31,Default,,0000,0000,0000,,Where I've actually already\Nlost some of the points for
Dialogue: 0,0:01:57.31,0:02:00.14,Default,,0000,0000,0000,,this already, so I'll use\Nit if we sketch the graph.
Dialogue: 0,0:02:02.41,0:02:06.99,Default,,0000,0000,0000,,You can see that this function\Ngets closer and closer to 0.
Dialogue: 0,0:02:07.69,0:02:09.41,Default,,0000,0000,0000,,In fact, whichever tiny distance
Dialogue: 0,0:02:09.41,0:02:13.39,Default,,0000,0000,0000,,we choose. Like this distance\NI've already sketched in here.
Dialogue: 0,0:02:14.30,0:02:18.33,Default,,0000,0000,0000,,Whatever distance we choose this\Nfunction will get within that
Dialogue: 0,0:02:18.33,0:02:20.75,Default,,0000,0000,0000,,from zero and stay that close.
Dialogue: 0,0:02:21.65,0:02:25.54,Default,,0000,0000,0000,,If a function does this,\Nwe say it tends to limit
Dialogue: 0,0:02:25.54,0:02:26.96,Default,,0000,0000,0000,,O's extends to Infinity.
Dialogue: 0,0:02:28.07,0:02:32.39,Default,,0000,0000,0000,,If a function tends to zeros\Nextends to Infinity, we write it
Dialogue: 0,0:02:32.39,0:02:38.43,Default,,0000,0000,0000,,like this. Who writes FX\Ntends to 0?
Dialogue: 0,0:02:39.33,0:02:41.12,Default,,0000,0000,0000,,As X tends to Infinity.
Dialogue: 0,0:02:42.89,0:02:50.16,Default,,0000,0000,0000,,For sometimes the limits of FX\Nas X tends to Infinity equals 0.
Dialogue: 0,0:02:52.29,0:02:58.44,Default,,0000,0000,0000,,Here's another\Nfunction with
Dialogue: 0,0:02:58.44,0:03:01.52,Default,,0000,0000,0000,,the limit.
Dialogue: 0,0:03:03.00,0:03:05.32,Default,,0000,0000,0000,,This time will have F of X.
Dialogue: 0,0:03:05.97,0:03:12.01,Default,,0000,0000,0000,,Equals 3 - 1 over X squared,\N4X greater than 0.
Dialogue: 0,0:03:13.58,0:03:17.22,Default,,0000,0000,0000,,So I'll just sketch this graph.
Dialogue: 0,0:03:18.41,0:03:20.18,Default,,0000,0000,0000,,Will have three about here.
Dialogue: 0,0:03:29.04,0:03:31.35,Default,,0000,0000,0000,,Right?
Dialogue: 0,0:03:32.72,0:03:34.82,Default,,0000,0000,0000,,This graph looks\Nsomething like this.
Dialogue: 0,0:03:42.65,0:03:45.84,Default,,0000,0000,0000,,Now again, whatever tiny\Ndistance I choose around 3:00.
Dialogue: 0,0:03:48.45,0:03:50.39,Default,,0000,0000,0000,,It doesn't matter\Nhow small this is.
Dialogue: 0,0:03:51.99,0:03:53.06,Default,,0000,0000,0000,,This function.
Dialogue: 0,0:03:55.49,0:03:59.05,Default,,0000,0000,0000,,Eventually gets trapped that far\Naway from 3.
Dialogue: 0,0:03:59.89,0:04:05.47,Default,,0000,0000,0000,,So this function has limit\Nthree, and again we write FX
Dialogue: 0,0:04:05.47,0:04:06.99,Default,,0000,0000,0000,,tends to three.
Dialogue: 0,0:04:08.05,0:04:09.62,Default,,0000,0000,0000,,As X tends to Infinity.
Dialogue: 0,0:04:10.21,0:04:13.67,Default,,0000,0000,0000,,For the limit of FX.
Dialogue: 0,0:04:13.67,0:04:16.66,Default,,0000,0000,0000,,Equals 3 as X tends to Infinity.
Dialogue: 0,0:04:18.80,0:04:23.39,Default,,0000,0000,0000,,In general, we say that the\Nfunction has limit L as X tends
Dialogue: 0,0:04:23.39,0:04:26.57,Default,,0000,0000,0000,,to Infinity. If how the smaller\Ndistance he choose.
Dialogue: 0,0:04:27.19,0:04:31.28,Default,,0000,0000,0000,,The function gets closer than\Nthat and stays closer than that
Dialogue: 0,0:04:31.28,0:04:32.77,Default,,0000,0000,0000,,as X gets larger.
Dialogue: 0,0:04:35.03,0:04:42.10,Default,,0000,0000,0000,,Not all functions have\Nreal limits as X tends
Dialogue: 0,0:04:42.10,0:04:46.80,Default,,0000,0000,0000,,to Infinity. Here's one\Nthat doesn't.
Dialogue: 0,0:04:48.10,0:04:52.04,Default,,0000,0000,0000,,Will have F
Dialogue: 0,0:04:52.04,0:04:55.77,Default,,0000,0000,0000,,of X. Equals X squared.
Dialogue: 0,0:04:56.66,0:04:58.99,Default,,0000,0000,0000,,And that is, I'm sure, you\Nknow, looks something like
Dialogue: 0,0:04:58.99,0:04:59.22,Default,,0000,0000,0000,,this.
Dialogue: 0,0:05:05.50,0:05:08.95,Default,,0000,0000,0000,,Now as X gets larger, this\Ncertainly doesn't get any closer
Dialogue: 0,0:05:08.95,0:05:10.21,Default,,0000,0000,0000,,to any real number.
Dialogue: 0,0:05:11.38,0:05:14.21,Default,,0000,0000,0000,,In fact, however, large\Nnumber I choose.
Dialogue: 0,0:05:16.48,0:05:20.30,Default,,0000,0000,0000,,This function will eventually\Nget bigger and stay bigger than
Dialogue: 0,0:05:20.30,0:05:22.97,Default,,0000,0000,0000,,it as X goes off to Infinity.
Dialogue: 0,0:05:24.23,0:05:28.81,Default,,0000,0000,0000,,If a function does this, we say\Nthat it has limits Infinity as X
Dialogue: 0,0:05:28.81,0:05:35.38,Default,,0000,0000,0000,,tends to Infinity. We write F\Nof X tends to Infinity.
Dialogue: 0,0:05:36.29,0:05:37.98,Default,,0000,0000,0000,,As X tends to Infinity.
Dialogue: 0,0:05:38.49,0:05:45.10,Default,,0000,0000,0000,,Or limits. With X as\NX tends to Infinity equals
Dialogue: 0,0:05:45.10,0:05:52.18,Default,,0000,0000,0000,,Infinity. If we\Nhave a function like
Dialogue: 0,0:05:52.18,0:05:55.95,Default,,0000,0000,0000,,F of X equals
Dialogue: 0,0:05:55.95,0:06:02.04,Default,,0000,0000,0000,,minus X. It\Nlooks like this.
Dialogue: 0,0:06:04.30,0:06:11.86,Default,,0000,0000,0000,,This also doesn't tend to a\Nreal limit as X tends to
Dialogue: 0,0:06:11.86,0:06:15.71,Default,,0000,0000,0000,,Infinity. But it doesn't tend to\NInfinity either, because it
Dialogue: 0,0:06:15.71,0:06:16.80,Default,,0000,0000,0000,,doesn't get very large.
Dialogue: 0,0:06:17.51,0:06:19.38,Default,,0000,0000,0000,,In fact, have a large and
Dialogue: 0,0:06:19.38,0:06:21.46,Default,,0000,0000,0000,,negative. A number I choose.
Dialogue: 0,0:06:22.67,0:06:26.31,Default,,0000,0000,0000,,This function will get below\Nthat number and stay below it.
Dialogue: 0,0:06:27.23,0:06:30.67,Default,,0000,0000,0000,,If a function does this, we'd\Nsay it tends to minus Infinity,
Dialogue: 0,0:06:30.67,0:06:32.11,Default,,0000,0000,0000,,as X tends to Infinity.
Dialogue: 0,0:06:33.19,0:06:36.32,Default,,0000,0000,0000,,So we write. F of X.
Dialogue: 0,0:06:36.87,0:06:38.56,Default,,0000,0000,0000,,Tends to minus Infinity.
Dialogue: 0,0:06:39.36,0:06:41.10,Default,,0000,0000,0000,,As X tends to Infinity.
Dialogue: 0,0:06:41.83,0:06:45.15,Default,,0000,0000,0000,,Oregon limit of F of X.
Dialogue: 0,0:06:45.98,0:06:49.76,Default,,0000,0000,0000,,Equals minus Infinity\Nas X tends to Infinity.
Dialogue: 0,0:06:50.91,0:06:56.04,Default,,0000,0000,0000,,Some functions that\Nhave any limits at all,
Dialogue: 0,0:06:56.04,0:06:59.24,Default,,0000,0000,0000,,as X tends to Infinity.
Dialogue: 0,0:07:00.33,0:07:06.76,Default,,0000,0000,0000,,This is the graph FX\Nequals X sign X.
Dialogue: 0,0:07:08.17,0:07:11.37,Default,,0000,0000,0000,,But as you can see, it certainly\Ndoesn't get closer to any real
Dialogue: 0,0:07:11.37,0:07:13.59,Default,,0000,0000,0000,,number. As extends to Infinity.
Dialogue: 0,0:07:15.09,0:07:17.87,Default,,0000,0000,0000,,Also, if I pick a really\Nlarge number.
Dialogue: 0,0:07:21.31,0:07:22.84,Default,,0000,0000,0000,,The graph will eventually get
Dialogue: 0,0:07:22.84,0:07:26.00,Default,,0000,0000,0000,,above it. But it won't stay\Nabove it because it always
Dialogue: 0,0:07:26.00,0:07:27.73,Default,,0000,0000,0000,,come back down to zero and\Ngo negative.
Dialogue: 0,0:07:28.77,0:07:31.63,Default,,0000,0000,0000,,Also, for pick a really large\Nnegative number, it will go
Dialogue: 0,0:07:31.63,0:07:34.49,Default,,0000,0000,0000,,below it, but it won't stay\Nbelow it because it'll go
Dialogue: 0,0:07:34.49,0:07:36.83,Default,,0000,0000,0000,,back up to zero and then go\Npositive again.
Dialogue: 0,0:07:37.99,0:07:41.78,Default,,0000,0000,0000,,So this function doesn't tend to\NInfinity, and it doesn't tend to
Dialogue: 0,0:07:41.78,0:07:43.36,Default,,0000,0000,0000,,in minus Infinity as well.
Dialogue: 0,0:07:44.29,0:07:46.46,Default,,0000,0000,0000,,So this function doesn't\Nhave any limited tool.
Dialogue: 0,0:07:47.80,0:07:54.60,Default,,0000,0000,0000,,We can also define limits\Nfor functions as X tends to
Dialogue: 0,0:07:54.60,0:07:55.83,Default,,0000,0000,0000,,minus Infinity.
Dialogue: 0,0:07:57.46,0:08:01.14,Default,,0000,0000,0000,,Here's a graph of E to the X.
Dialogue: 0,0:08:03.16,0:08:06.65,Default,,0000,0000,0000,,So. This after 0 gets
Dialogue: 0,0:08:06.65,0:08:09.17,Default,,0000,0000,0000,,large. The 40
Dialogue: 0,0:08:10.41,0:08:12.49,Default,,0000,0000,0000,,Gets closer and closer to 0.
Dialogue: 0,0:08:15.41,0:08:19.39,Default,,0000,0000,0000,,As X gets more negative here.
Dialogue: 0,0:08:19.94,0:08:22.94,Default,,0000,0000,0000,,This function gets closer and\Ncloser to 0.
Dialogue: 0,0:08:24.32,0:08:27.45,Default,,0000,0000,0000,,And however small a\Ndistance I choose.
Dialogue: 0,0:08:29.32,0:08:35.28,Default,,0000,0000,0000,,The function will get closer\Nthan that to zero and stay
Dialogue: 0,0:08:35.28,0:08:39.08,Default,,0000,0000,0000,,closer than that as X gets more
Dialogue: 0,0:08:39.08,0:08:42.72,Default,,0000,0000,0000,,negative. If a function is this
Dialogue: 0,0:08:42.72,0:08:46.61,Default,,0000,0000,0000,,we say? F of X tends
Dialogue: 0,0:08:46.61,0:08:52.80,Default,,0000,0000,0000,,to. 0. As\NX tends to minus Infinity.
Dialogue: 0,0:08:53.85,0:09:00.17,Default,,0000,0000,0000,,Oh, limit as X tends to minus\NInfinity of F of X.
Dialogue: 0,0:09:00.86,0:09:01.85,Default,,0000,0000,0000,,Equals 0.
Dialogue: 0,0:09:03.59,0:09:08.68,Default,,0000,0000,0000,,In general, we say a function\Nhas limit L's extends to minus
Dialogue: 0,0:09:08.68,0:09:12.80,Default,,0000,0000,0000,,Infinity. If how the smaller\Ndistance I choose the function
Dialogue: 0,0:09:12.80,0:09:17.26,Default,,0000,0000,0000,,gets closer than that and stays\Ncloser than that as X gets more
Dialogue: 0,0:09:17.26,0:09:22.74,Default,,0000,0000,0000,,negative. If we\Nhave a graph
Dialogue: 0,0:09:22.74,0:09:25.43,Default,,0000,0000,0000,,like X squared.
Dialogue: 0,0:09:26.62,0:09:28.56,Default,,0000,0000,0000,,Shoulder sketch.
Dialogue: 0,0:09:34.39,0:09:41.25,Default,,0000,0000,0000,,You can see that this doesn't\Nget closer to any real number as
Dialogue: 0,0:09:41.25,0:09:43.89,Default,,0000,0000,0000,,X goes to minus Infinity.
Dialogue: 0,0:09:44.97,0:09:47.30,Default,,0000,0000,0000,,But however larger\Nnumber I choose.
Dialogue: 0,0:09:50.03,0:09:51.76,Default,,0000,0000,0000,,As X gets more negative.
Dialogue: 0,0:09:52.34,0:09:55.58,Default,,0000,0000,0000,,This graph will go above\Nthat number and stay above
Dialogue: 0,0:09:55.58,0:09:56.23,Default,,0000,0000,0000,,that number.
Dialogue: 0,0:09:57.66,0:10:02.07,Default,,0000,0000,0000,,If a function does this, we say\Nit tends to Infinity as X tends
Dialogue: 0,0:10:02.07,0:10:08.07,Default,,0000,0000,0000,,to minus Infinity. So F of\NX tends to Infinity.
Dialogue: 0,0:10:08.94,0:10:12.82,Default,,0000,0000,0000,,As X tends to minus Infinity.
Dialogue: 0,0:10:12.82,0:10:19.10,Default,,0000,0000,0000,,Limits of F of X equals Infinity\Nas X tends to minus Infinity.
Dialogue: 0,0:10:19.87,0:10:26.89,Default,,0000,0000,0000,,Now function like X\Ncubed doesn't do either
Dialogue: 0,0:10:26.89,0:10:33.90,Default,,0000,0000,0000,,of these things. The\Ngraph is something like
Dialogue: 0,0:10:33.90,0:10:34.78,Default,,0000,0000,0000,,this.
Dialogue: 0,0:10:36.34,0:10:39.52,Default,,0000,0000,0000,,So it certainly doesn't get\Nlarger than any number I pick as
Dialogue: 0,0:10:39.52,0:10:42.17,Default,,0000,0000,0000,,X goes to minus Infinity and\Nstay larger than it.
Dialogue: 0,0:10:42.71,0:10:45.54,Default,,0000,0000,0000,,And it doesn't get closer to any\Nreal number, so it doesn't have
Dialogue: 0,0:10:45.54,0:10:50.22,Default,,0000,0000,0000,,a real limit. But however larger\Nnegative number I choose.
Dialogue: 0,0:10:52.24,0:10:55.60,Default,,0000,0000,0000,,As X gets more negative, this\Nfunction will drop below that
Dialogue: 0,0:10:55.60,0:10:59.56,Default,,0000,0000,0000,,number and stay below it and\Nthis works for any number I can
Dialogue: 0,0:10:59.56,0:11:06.01,Default,,0000,0000,0000,,choose. When this happens, we\Nsay F of X tends to minus
Dialogue: 0,0:11:06.01,0:11:09.08,Default,,0000,0000,0000,,Infinity. As X tends to minus
Dialogue: 0,0:11:09.08,0:11:13.27,Default,,0000,0000,0000,,Infinity. For, again, the limits\Nof F of X.
Dialogue: 0,0:11:13.98,0:11:15.61,Default,,0000,0000,0000,,Equals minus Infinity.
Dialogue: 0,0:11:16.14,0:11:18.12,Default,,0000,0000,0000,,As X tends to minus Infinity.
Dialogue: 0,0:11:20.13,0:11:25.60,Default,,0000,0000,0000,,Now again, a function didn't\Nhave any limit as X tends to
Dialogue: 0,0:11:25.60,0:11:29.84,Default,,0000,0000,0000,,minus Infinity. And our graph\Nof FX equals X sign X is a
Dialogue: 0,0:11:29.84,0:11:31.02,Default,,0000,0000,0000,,good example of this again.
Dialogue: 0,0:11:32.25,0:11:35.34,Default,,0000,0000,0000,,As X gets more negative, this\Ncertainly doesn't get closer to
Dialogue: 0,0:11:35.34,0:11:38.15,Default,,0000,0000,0000,,any real number, so it doesn't\Nhave a real limit.
Dialogue: 0,0:11:38.71,0:11:41.82,Default,,0000,0000,0000,,It might go above any number we\Ncan pick, but it won't stay
Dialogue: 0,0:11:41.82,0:11:44.45,Default,,0000,0000,0000,,above it because it comes back\Ndown and goes through zero.
Dialogue: 0,0:11:45.13,0:11:48.51,Default,,0000,0000,0000,,And it won't get more. It won't\Nget more and stay more negative
Dialogue: 0,0:11:48.51,0:11:51.11,Default,,0000,0000,0000,,than any large negative number\Nwe can choose, because always
Dialogue: 0,0:11:51.11,0:11:52.93,Default,,0000,0000,0000,,come back up to zero and go
Dialogue: 0,0:11:52.93,0:11:57.15,Default,,0000,0000,0000,,positive again. So this is a\Nfunction that has no limit as X
Dialogue: 0,0:11:57.15,0:11:58.20,Default,,0000,0000,0000,,tends to minus Infinity.
Dialogue: 0,0:11:59.54,0:12:06.03,Default,,0000,0000,0000,,There's one more type\Nof limit. We can
Dialogue: 0,0:12:06.03,0:12:10.41,Default,,0000,0000,0000,,define functions. Let's\Nlook at the graph of the
Dialogue: 0,0:12:10.41,0:12:13.42,Default,,0000,0000,0000,,really simple function\Nlike F of X equals X +3.
Dialogue: 0,0:12:15.87,0:12:18.96,Default,,0000,0000,0000,,So that looks something like
Dialogue: 0,0:12:18.96,0:12:20.39,Default,,0000,0000,0000,,this. Three there.
Dialogue: 0,0:12:21.76,0:12:28.46,Default,,0000,0000,0000,,If I pick a\Npoint on this graph,
Dialogue: 0,0:12:28.46,0:12:30.13,Default,,0000,0000,0000,,like say.
Dialogue: 0,0:12:31.59,0:12:33.00,Default,,0000,0000,0000,,Here, when X is one.
Dialogue: 0,0:12:35.74,0:12:37.50,Default,,0000,0000,0000,,The graph goes through 1 four.
Dialogue: 0,0:12:39.81,0:12:42.34,Default,,0000,0000,0000,,You can see that as X approaches
Dialogue: 0,0:12:42.34,0:12:45.100,Default,,0000,0000,0000,,one. The value of the\Nfunction approaches 4.
Dialogue: 0,0:12:47.85,0:12:50.31,Default,,0000,0000,0000,,We say that.
Dialogue: 0,0:12:50.31,0:12:52.83,Default,,0000,0000,0000,,F of X the limit of F of X.
Dialogue: 0,0:12:53.48,0:12:56.78,Default,,0000,0000,0000,,Equals 4 as X tends to one.
Dialogue: 0,0:12:58.37,0:12:59.23,Default,,0000,0000,0000,,Similarly.
Dialogue: 0,0:13:00.47,0:13:02.96,Default,,0000,0000,0000,,If we say X equals 5.
Dialogue: 0,0:13:04.37,0:13:06.21,Default,,0000,0000,0000,,Then the function.
Dialogue: 0,0:13:07.55,0:13:12.65,Default,,0000,0000,0000,,Is it 8? And his ex gets closer\Nto five. The function gets close
Dialogue: 0,0:13:12.65,0:13:19.61,Default,,0000,0000,0000,,to 8. So the limit of F\Nof X as X tends to 5
Dialogue: 0,0:13:19.61,0:13:20.56,Default,,0000,0000,0000,,equals 8.
Dialogue: 0,0:13:22.55,0:13:24.00,Default,,0000,0000,0000,,Now this doesn't look very
Dialogue: 0,0:13:24.00,0:13:27.70,Default,,0000,0000,0000,,useful. But it does become\Nuseful if we have functions that
Dialogue: 0,0:13:27.70,0:13:28.86,Default,,0000,0000,0000,,aren't defined data point.
Dialogue: 0,0:13:29.88,0:13:36.64,Default,,0000,0000,0000,,This is the case with\Nthis function. This is F
Dialogue: 0,0:13:36.64,0:13:39.69,Default,,0000,0000,0000,,of X. Equals.
Dialogue: 0,0:13:40.23,0:13:43.33,Default,,0000,0000,0000,,X sign one over X.
Dialogue: 0,0:13:45.20,0:13:48.90,Default,,0000,0000,0000,,You can see for this function\Nthat as X gets close to 0.
Dialogue: 0,0:13:49.68,0:13:50.96,Default,,0000,0000,0000,,The function gets very very
Dialogue: 0,0:13:50.96,0:13:55.98,Default,,0000,0000,0000,,close to 0. But the cause at X\Nequals 0. We have a zero
Dialogue: 0,0:13:55.98,0:13:59.64,Default,,0000,0000,0000,,denominator. This function isn't\Ndefined for X equals 0.
Dialogue: 0,0:14:01.51,0:14:04.82,Default,,0000,0000,0000,,But because the function gets\Ncloser and closer to zero, it's
Dialogue: 0,0:14:04.82,0:14:08.73,Default,,0000,0000,0000,,almost as if the value of the\Nfunction at zero should be 0.
Dialogue: 0,0:14:10.78,0:14:14.43,Default,,0000,0000,0000,,So the limit of F of X.
Dialogue: 0,0:14:15.33,0:14:18.76,Default,,0000,0000,0000,,As X tends to 0 equals 0.
Dialogue: 0,0:14:19.33,0:14:20.82,Default,,0000,0000,0000,,And you can think of 0.
Dialogue: 0,0:14:21.34,0:14:25.29,Default,,0000,0000,0000,,As the value F of X should\Ntake when X equals 0, even
Dialogue: 0,0:14:25.29,0:14:26.51,Default,,0000,0000,0000,,though it's not defined.
Dialogue: 0,0:14:28.03,0:14:30.96,Default,,0000,0000,0000,,Here's another example.
Dialogue: 0,0:14:30.96,0:14:37.55,Default,,0000,0000,0000,,This function is F\Nof X equals Y
Dialogue: 0,0:14:37.55,0:14:43.32,Default,,0000,0000,0000,,to the minus one\Nover X squared.
Dialogue: 0,0:14:44.52,0:14:48.58,Default,,0000,0000,0000,,Now again, when X equals 0, we\Nhave a zero denominator. Here in
Dialogue: 0,0:14:48.58,0:14:52.10,Default,,0000,0000,0000,,the function. So it's not\Ndefined at X equals 0.
Dialogue: 0,0:14:53.34,0:14:56.31,Default,,0000,0000,0000,,But as X gets close to 0.
Dialogue: 0,0:14:56.96,0:15:02.46,Default,,0000,0000,0000,,This function approaches 0, so\Nagain the limit of F of X as X
Dialogue: 0,0:15:02.46,0:15:07.96,Default,,0000,0000,0000,,tends to zero in this case is\Nzero, and again we can think of
Dialogue: 0,0:15:07.96,0:15:12.29,Default,,0000,0000,0000,,0. Is the value F should take\Nwhen X equals 0.
Dialogue: 0,0:15:14.36,0:15:21.51,Default,,0000,0000,0000,,Not all functions\Nhave nice limits
Dialogue: 0,0:15:21.51,0:15:23.90,Default,,0000,0000,0000,,like this.
Dialogue: 0,0:15:25.09,0:15:31.97,Default,,0000,0000,0000,,If we look at the function\NMoD X over XF of X
Dialogue: 0,0:15:31.97,0:15:34.83,Default,,0000,0000,0000,,equals MoD X over X.
Dialogue: 0,0:15:36.51,0:15:38.07,Default,,0000,0000,0000,,If X is positive.
Dialogue: 0,0:15:39.02,0:15:44.58,Default,,0000,0000,0000,,Then The\Ntop of this
Dialogue: 0,0:15:44.58,0:15:46.88,Default,,0000,0000,0000,,is just X.
Dialogue: 0,0:15:48.09,0:15:51.88,Default,,0000,0000,0000,,So this becomes X over X,\Nwhich is one.
Dialogue: 0,0:15:52.96,0:15:55.21,Default,,0000,0000,0000,,But if X is less than 0.
Dialogue: 0,0:15:55.84,0:16:01.04,Default,,0000,0000,0000,,This becomes little becomes\Nminus X, so that's minus X over
Dialogue: 0,0:16:01.04,0:16:03.51,Default,,0000,0000,0000,,X. So that's minus one.
Dialogue: 0,0:16:04.55,0:16:08.69,Default,,0000,0000,0000,,When X is 0, is not defined at\Nall, so the graph looks like
Dialogue: 0,0:16:08.69,0:16:10.90,Default,,0000,0000,0000,,this. It's just one.
Dialogue: 0,0:16:12.48,0:16:14.57,Default,,0000,0000,0000,,When X is greater than zero and
Dialogue: 0,0:16:14.57,0:16:16.90,Default,,0000,0000,0000,,it's just. Minus one.
Dialogue: 0,0:16:19.65,0:16:21.01,Default,,0000,0000,0000,,When X is less than 0.
Dialogue: 0,0:16:22.59,0:16:26.30,Default,,0000,0000,0000,,And you can see here there's no\Nway we can define a good limit
Dialogue: 0,0:16:26.30,0:16:29.48,Default,,0000,0000,0000,,for this, because as we approach\Nzero from the right hand side.
Dialogue: 0,0:16:30.10,0:16:33.88,Default,,0000,0000,0000,,It looks like it should be one,\Nbut if we approach zero from the
Dialogue: 0,0:16:33.88,0:16:36.85,Default,,0000,0000,0000,,left hand side, the function\Nlooks like it should be taking
Dialogue: 0,0:16:36.85,0:16:41.25,Default,,0000,0000,0000,,value minus one. So here it's\Nnot clear what ever vex should
Dialogue: 0,0:16:41.25,0:16:42.84,Default,,0000,0000,0000,,be when X is 0.
Dialogue: 0,0:16:43.39,0:16:46.35,Default,,0000,0000,0000,,Because there isn't a proper\Nlimit for this function as X
Dialogue: 0,0:16:46.35,0:16:49.85,Default,,0000,0000,0000,,tends to 0, so this is a good\Nexample of a function that
Dialogue: 0,0:16:49.85,0:16:52.54,Default,,0000,0000,0000,,doesn't have a limit as X\Ntends to a number.