0:00:01.430,0:00:05.265 In this video, we'll define what[br]it means for a function F to 0:00:05.265,0:00:07.625 tend to a limit as X tends to 0:00:07.625,0:00:12.085 Infinity. We also define what it[br]means for its tend to limit as X 0:00:12.085,0:00:15.025 tends to minus Infinity, and as[br]X tends to a real number. 0:00:17.440,0:00:21.059 Remember from the video limits[br]of sequences, we define what it 0:00:21.059,0:00:25.994 meant for a sequence to tend to[br]A to a limit as N tends to 0:00:25.994,0:00:32.274 Infinity, we said. Sequence[br]wired tends to limit L 0:00:32.274,0:00:36.169 as an tends to Infinity. 0:00:37.190,0:00:40.112 If however, small a number I 0:00:40.112,0:00:46.504 chose. The sequence YN would get[br]that close to L and stay that 0:00:46.504,0:00:51.957 close. One of our examples[br]was the sequence YN equals 0:00:51.957,0:00:53.346 one over N. 0:00:54.480,0:00:56.916 I just put a graph of this. 0:01:01.720,0:01:05.440 So her plot the points[br]of why and which was 0:01:05.440,0:01:07.300 one equals one over N. 0:01:08.400,0:01:11.920 It looks like this sort of it 1. 0:01:12.500,0:01:16.236 And then two was a half. Then it 0:01:16.236,0:01:23.155 went 1/3. Now this[br]sequence tended to 0:01:23.155,0:01:26.380 0 because eventually. 0:01:26.930,0:01:29.390 The points in the sequence get 0:01:29.390,0:01:34.844 as close. Zeros alike, so have a[br]smaller distance I choose. 0:01:38.010,0:01:41.475 The points in the sequence[br]will eventually get within 0:01:41.475,0:01:44.555 that boundary from zero and[br]stay in there. 0:01:45.710,0:01:48.900 The functions are defined[br]in a similar way to limits 0:01:48.900,0:01:49.538 of sequences. 0:01:50.950,0:01:54.184 If we have a function like FX[br]equals one over X. 0:01:54.740,0:01:57.310 Where I've actually already[br]lost some of the points for 0:01:57.310,0:02:00.137 this already, so I'll use[br]it if we sketch the graph. 0:02:02.410,0:02:06.994 You can see that this function[br]gets closer and closer to 0. 0:02:07.690,0:02:09.410 In fact, whichever tiny distance 0:02:09.410,0:02:13.390 we choose. Like this distance[br]I've already sketched in here. 0:02:14.300,0:02:18.330 Whatever distance we choose this[br]function will get within that 0:02:18.330,0:02:20.748 from zero and stay that close. 0:02:21.650,0:02:25.544 If a function does this,[br]we say it tends to limit 0:02:25.544,0:02:26.960 O's extends to Infinity. 0:02:28.070,0:02:32.390 If a function tends to zeros[br]extends to Infinity, we write it 0:02:32.390,0:02:38.426 like this. Who writes FX[br]tends to 0? 0:02:39.330,0:02:41.120 As X tends to Infinity. 0:02:42.890,0:02:50.157 For sometimes the limits of FX[br]as X tends to Infinity equals 0. 0:02:52.290,0:02:58.442 Here's another[br]function with 0:02:58.442,0:03:01.518 the limit. 0:03:03.000,0:03:05.324 This time will have F of X. 0:03:05.970,0:03:12.009 Equals 3 - 1 over X squared,[br]4X greater than 0. 0:03:13.580,0:03:17.216 So I'll just sketch this graph. 0:03:18.410,0:03:20.180 Will have three about here. 0:03:29.040,0:03:31.350 Right? 0:03:32.720,0:03:34.820 This graph looks[br]something like this. 0:03:42.650,0:03:45.836 Now again, whatever tiny[br]distance I choose around 3:00. 0:03:48.450,0:03:50.389 It doesn't matter[br]how small this is. 0:03:51.990,0:03:53.060 This function. 0:03:55.490,0:03:59.050 Eventually gets trapped that far[br]away from 3. 0:03:59.890,0:04:05.467 So this function has limit[br]three, and again we write FX 0:04:05.467,0:04:06.988 tends to three. 0:04:08.050,0:04:09.620 As X tends to Infinity. 0:04:10.210,0:04:13.670 For the limit of FX. 0:04:13.670,0:04:16.659 Equals 3 as X tends to Infinity. 0:04:18.800,0:04:23.389 In general, we say that the[br]function has limit L as X tends 0:04:23.389,0:04:26.566 to Infinity. If how the smaller[br]distance he choose. 0:04:27.190,0:04:31.282 The function gets closer than[br]that and stays closer than that 0:04:31.282,0:04:32.770 as X gets larger. 0:04:35.030,0:04:42.095 Not all functions have[br]real limits as X tends 0:04:42.095,0:04:46.805 to Infinity. Here's one[br]that doesn't. 0:04:48.100,0:04:52.042 Will have F 0:04:52.042,0:04:55.768 of X. Equals X squared. 0:04:56.660,0:04:58.990 And that is, I'm sure, you[br]know, looks something like 0:04:58.990,0:04:59.223 this. 0:05:05.500,0:05:08.954 Now as X gets larger, this[br]certainly doesn't get any closer 0:05:08.954,0:05:10.210 to any real number. 0:05:11.380,0:05:14.208 In fact, however, large[br]number I choose. 0:05:16.480,0:05:20.300 This function will eventually[br]get bigger and stay bigger than 0:05:20.300,0:05:22.974 it as X goes off to Infinity. 0:05:24.230,0:05:28.808 If a function does this, we say[br]that it has limits Infinity as X 0:05:28.808,0:05:35.384 tends to Infinity. We write F[br]of X tends to Infinity. 0:05:36.290,0:05:37.980 As X tends to Infinity. 0:05:38.490,0:05:45.096 Or limits. With X as[br]X tends to Infinity equals 0:05:45.096,0:05:52.178 Infinity. If we[br]have a function like 0:05:52.178,0:05:55.950 F of X equals 0:05:55.950,0:06:02.038 minus X. It[br]looks like this. 0:06:04.300,0:06:11.860 This also doesn't tend to a[br]real limit as X tends to 0:06:11.860,0:06:15.707 Infinity. But it doesn't tend to[br]Infinity either, because it 0:06:15.707,0:06:16.799 doesn't get very large. 0:06:17.510,0:06:19.376 In fact, have a large and 0:06:19.376,0:06:21.458 negative. A number I choose. 0:06:22.670,0:06:26.311 This function will get below[br]that number and stay below it. 0:06:27.230,0:06:30.674 If a function does this, we'd[br]say it tends to minus Infinity, 0:06:30.674,0:06:32.109 as X tends to Infinity. 0:06:33.190,0:06:36.320 So we write. F of X. 0:06:36.870,0:06:38.558 Tends to minus Infinity. 0:06:39.360,0:06:41.100 As X tends to Infinity. 0:06:41.830,0:06:45.148 Oregon limit of F of X. 0:06:45.980,0:06:49.756 Equals minus Infinity[br]as X tends to Infinity. 0:06:50.910,0:06:56.038 Some functions that[br]have any limits at all, 0:06:56.038,0:06:59.243 as X tends to Infinity. 0:07:00.330,0:07:06.765 This is the graph FX[br]equals X sign X. 0:07:08.170,0:07:11.368 But as you can see, it certainly[br]doesn't get closer to any real 0:07:11.368,0:07:13.588 number. As extends to Infinity. 0:07:15.090,0:07:17.866 Also, if I pick a really[br]large number. 0:07:21.310,0:07:22.835 The graph will eventually get 0:07:22.835,0:07:26.004 above it. But it won't stay[br]above it because it always 0:07:26.004,0:07:27.732 come back down to zero and[br]go negative. 0:07:28.770,0:07:31.630 Also, for pick a really large[br]negative number, it will go 0:07:31.630,0:07:34.490 below it, but it won't stay[br]below it because it'll go 0:07:34.490,0:07:36.830 back up to zero and then go[br]positive again. 0:07:37.990,0:07:41.782 So this function doesn't tend to[br]Infinity, and it doesn't tend to 0:07:41.782,0:07:43.362 in minus Infinity as well. 0:07:44.290,0:07:46.458 So this function doesn't[br]have any limited tool. 0:07:47.800,0:07:54.598 We can also define limits[br]for functions as X tends to 0:07:54.598,0:07:55.834 minus Infinity. 0:07:57.460,0:08:01.140 Here's a graph of E to the X. 0:08:03.160,0:08:06.652 So. This after 0 gets 0:08:06.652,0:08:09.170 large. The 40 0:08:10.410,0:08:12.486 Gets closer and closer to 0. 0:08:15.410,0:08:19.388 As X gets more negative here. 0:08:19.940,0:08:22.940 This function gets closer and[br]closer to 0. 0:08:24.320,0:08:27.449 And however small a[br]distance I choose. 0:08:29.320,0:08:35.282 The function will get closer[br]than that to zero and stay 0:08:35.282,0:08:39.076 closer than that as X gets more 0:08:39.076,0:08:42.720 negative. If a function is this 0:08:42.720,0:08:46.614 we say? F of X tends 0:08:46.614,0:08:52.796 to. 0. As[br]X tends to minus Infinity. 0:08:53.850,0:09:00.174 Oh, limit as X tends to minus[br]Infinity of F of X. 0:09:00.860,0:09:01.850 Equals 0. 0:09:03.590,0:09:08.678 In general, we say a function[br]has limit L's extends to minus 0:09:08.678,0:09:12.797 Infinity. If how the smaller[br]distance I choose the function 0:09:12.797,0:09:17.256 gets closer than that and stays[br]closer than that as X gets more 0:09:17.256,0:09:22.745 negative. If we[br]have a graph 0:09:22.745,0:09:25.430 like X squared. 0:09:26.620,0:09:28.560 Shoulder sketch. 0:09:34.390,0:09:41.254 You can see that this doesn't[br]get closer to any real number as 0:09:41.254,0:09:43.894 X goes to minus Infinity. 0:09:44.970,0:09:47.298 But however larger[br]number I choose. 0:09:50.030,0:09:51.760 As X gets more negative. 0:09:52.340,0:09:55.580 This graph will go above[br]that number and stay above 0:09:55.580,0:09:56.228 that number. 0:09:57.660,0:10:02.070 If a function does this, we say[br]it tends to Infinity as X tends 0:10:02.070,0:10:08.074 to minus Infinity. So F of[br]X tends to Infinity. 0:10:08.940,0:10:12.816 As X tends to minus Infinity. 0:10:12.820,0:10:19.099 Limits of F of X equals Infinity[br]as X tends to minus Infinity. 0:10:19.870,0:10:26.886 Now function like X[br]cubed doesn't do either 0:10:26.886,0:10:33.902 of these things. The[br]graph is something like 0:10:33.902,0:10:34.779 this. 0:10:36.340,0:10:39.520 So it certainly doesn't get[br]larger than any number I pick as 0:10:39.520,0:10:42.170 X goes to minus Infinity and[br]stay larger than it. 0:10:42.710,0:10:45.544 And it doesn't get closer to any[br]real number, so it doesn't have 0:10:45.544,0:10:50.225 a real limit. But however larger[br]negative number I choose. 0:10:52.240,0:10:55.595 As X gets more negative, this[br]function will drop below that 0:10:55.595,0:10:59.560 number and stay below it and[br]this works for any number I can 0:10:59.560,0:11:06.010 choose. When this happens, we[br]say F of X tends to minus 0:11:06.010,0:11:09.080 Infinity. As X tends to minus 0:11:09.080,0:11:13.270 Infinity. For, again, the limits[br]of F of X. 0:11:13.980,0:11:15.609 Equals minus Infinity. 0:11:16.140,0:11:18.120 As X tends to minus Infinity. 0:11:20.130,0:11:25.602 Now again, a function didn't[br]have any limit as X tends to 0:11:25.602,0:11:29.836 minus Infinity. And our graph[br]of FX equals X sign X is a 0:11:29.836,0:11:31.016 good example of this again. 0:11:32.250,0:11:35.341 As X gets more negative, this[br]certainly doesn't get closer to 0:11:35.341,0:11:38.151 any real number, so it doesn't[br]have a real limit. 0:11:38.710,0:11:41.817 It might go above any number we[br]can pick, but it won't stay 0:11:41.817,0:11:44.446 above it because it comes back[br]down and goes through zero. 0:11:45.130,0:11:48.510 And it won't get more. It won't[br]get more and stay more negative 0:11:48.510,0:11:51.110 than any large negative number[br]we can choose, because always 0:11:51.110,0:11:52.930 come back up to zero and go 0:11:52.930,0:11:57.151 positive again. So this is a[br]function that has no limit as X 0:11:57.151,0:11:58.195 tends to minus Infinity. 0:11:59.540,0:12:06.028 There's one more type[br]of limit. We can 0:12:06.028,0:12:10.407 define functions. Let's[br]look at the graph of the 0:12:10.407,0:12:13.417 really simple function[br]like F of X equals X +3. 0:12:15.870,0:12:18.960 So that looks something like 0:12:18.960,0:12:20.390 this. Three there. 0:12:21.760,0:12:28.456 If I pick a[br]point on this graph, 0:12:28.456,0:12:30.130 like say. 0:12:31.590,0:12:33.000 Here, when X is one. 0:12:35.740,0:12:37.498 The graph goes through 1 four. 0:12:39.810,0:12:42.337 You can see that as X approaches 0:12:42.337,0:12:45.998 one. The value of the[br]function approaches 4. 0:12:47.850,0:12:50.310 We say that. 0:12:50.310,0:12:52.830 F of X the limit of F of X. 0:12:53.480,0:12:56.777 Equals 4 as X tends to one. 0:12:58.370,0:12:59.230 Similarly. 0:13:00.470,0:13:02.960 If we say X equals 5. 0:13:04.370,0:13:06.209 Then the function. 0:13:07.550,0:13:12.647 Is it 8? And his ex gets closer[br]to five. The function gets close 0:13:12.647,0:13:19.612 to 8. So the limit of F[br]of X as X tends to 5 0:13:19.612,0:13:20.564 equals 8. 0:13:22.550,0:13:24.000 Now this doesn't look very 0:13:24.000,0:13:27.700 useful. But it does become[br]useful if we have functions that 0:13:27.700,0:13:28.856 aren't defined data point. 0:13:29.880,0:13:36.640 This is the case with[br]this function. This is F 0:13:36.640,0:13:39.690 of X. Equals. 0:13:40.230,0:13:43.330 X sign one over X. 0:13:45.200,0:13:48.905 You can see for this function[br]that as X gets close to 0. 0:13:49.680,0:13:50.960 The function gets very very 0:13:50.960,0:13:55.980 close to 0. But the cause at X[br]equals 0. We have a zero 0:13:55.980,0:13:59.644 denominator. This function isn't[br]defined for X equals 0. 0:14:01.510,0:14:04.821 But because the function gets[br]closer and closer to zero, it's 0:14:04.821,0:14:08.734 almost as if the value of the[br]function at zero should be 0. 0:14:10.780,0:14:14.434 So the limit of F of X. 0:14:15.330,0:14:18.760 As X tends to 0 equals 0. 0:14:19.330,0:14:20.818 And you can think of 0. 0:14:21.340,0:14:25.292 As the value F of X should[br]take when X equals 0, even 0:14:25.292,0:14:26.508 though it's not defined. 0:14:28.030,0:14:30.958 Here's another example. 0:14:30.960,0:14:37.552 This function is F[br]of X equals Y 0:14:37.552,0:14:43.320 to the minus one[br]over X squared. 0:14:44.520,0:14:48.576 Now again, when X equals 0, we[br]have a zero denominator. Here in 0:14:48.576,0:14:52.098 the function. So it's not[br]defined at X equals 0. 0:14:53.340,0:14:56.308 But as X gets close to 0. 0:14:56.960,0:15:02.462 This function approaches 0, so[br]again the limit of F of X as X 0:15:02.462,0:15:07.964 tends to zero in this case is[br]zero, and again we can think of 0:15:07.964,0:15:12.287 0. Is the value F should take[br]when X equals 0. 0:15:14.360,0:15:21.512 Not all functions[br]have nice limits 0:15:21.512,0:15:23.896 like this. 0:15:25.090,0:15:31.966 If we look at the function[br]MoD X over XF of X 0:15:31.966,0:15:34.831 equals MoD X over X. 0:15:36.510,0:15:38.070 If X is positive. 0:15:39.020,0:15:44.578 Then The[br]top of this 0:15:44.578,0:15:46.879 is just X. 0:15:48.090,0:15:51.879 So this becomes X over X,[br]which is one. 0:15:52.960,0:15:55.214 But if X is less than 0. 0:15:55.840,0:16:01.043 This becomes little becomes[br]minus X, so that's minus X over 0:16:01.043,0:16:03.510 X. So that's minus one. 0:16:04.550,0:16:08.694 When X is 0, is not defined at[br]all, so the graph looks like 0:16:08.694,0:16:10.898 this. It's just one. 0:16:12.480,0:16:14.566 When X is greater than zero and 0:16:14.566,0:16:16.900 it's just. Minus one. 0:16:19.650,0:16:21.006 When X is less than 0. 0:16:22.590,0:16:26.300 And you can see here there's no[br]way we can define a good limit 0:16:26.300,0:16:29.480 for this, because as we approach[br]zero from the right hand side. 0:16:30.100,0:16:33.880 It looks like it should be one,[br]but if we approach zero from the 0:16:33.880,0:16:36.850 left hand side, the function[br]looks like it should be taking 0:16:36.850,0:16:41.252 value minus one. So here it's[br]not clear what ever vex should 0:16:41.252,0:16:42.842 be when X is 0. 0:16:43.390,0:16:46.349 Because there isn't a proper[br]limit for this function as X 0:16:46.349,0:16:49.846 tends to 0, so this is a good[br]example of a function that 0:16:49.846,0:16:52.536 doesn't have a limit as X[br]tends to a number.