1 00:00:00,000 --> 00:00:00,043 2 00:00:00,050 --> 00:00:01,076 Welcome back. 3 00:00:01,082 --> 00:00:04,043 Now that we hopefully have a little bit of an intuition 4 00:00:04,049 --> 00:00:07,059 of what a limit is or finding the limit of a function is. 5 00:00:07,067 --> 00:00:09,003 Let's do some problems. 6 00:00:09,009 --> 00:00:12,026 Some of these you might see on your exams or when you're 7 00:00:12,032 --> 00:00:14,070 actually trying to solve a general limit problem. 8 00:00:14,076 --> 00:00:19,009 So let's say that, what is the limit? 9 00:00:19,016 --> 00:00:23,016 Once again, my pen is not working. 10 00:00:23,022 --> 00:00:32,049 What is the limit as x approaches? 11 00:00:32,057 --> 00:00:37,000 Let's say, -1, and let me see, what's a good? 12 00:00:37,006 --> 00:00:42,070 Let's say my expression is-- I'll put parenthesis 13 00:00:42,076 --> 00:00:52,030 so it's cleaner, say it's (2x + 2)/(x + 1). 14 00:00:52,067 --> 00:00:56,009 So the first thing I would always try to do is just say, 15 00:00:56,017 --> 00:00:59,003 well, what happens if I just stick x straight into this expression? 16 00:00:59,003 --> 00:01:00,017 What happens? 17 00:01:00,017 --> 00:01:03,017 Well, what's 2x + 2 when x = -1? 18 00:01:03,017 --> 00:01:05,020 It's 2 times -1, 19 00:01:05,027 --> 00:01:11,073 (2 times -1 + 2)/(-1 + 1). 20 00:01:12,003 --> 00:01:16,033 Well, the numerator is -2 + 2, that equals 0, 21 00:01:16,040 --> 00:01:18,063 over, what's the denominator? 22 00:01:18,069 --> 00:01:21,019 (-1 + 1)/0. 23 00:01:21,027 --> 00:01:23,040 And do we know what 0/0 is? 24 00:01:23,046 --> 00:01:26,003 Well, well no, it's undefined, right? 25 00:01:26,010 --> 00:01:29,063 So here's a case, just like what we saw in that first video 26 00:01:29,069 --> 00:01:34,029 where the limit can't equal what the expression equals 27 00:01:34,037 --> 00:01:36,030 when you substitute x for the number 28 00:01:36,037 --> 00:01:38,080 you're trying to find the limit of because you get 29 00:01:38,087 --> 00:01:40,049 an undefined answer. 30 00:01:40,057 --> 00:01:42,090 So see if using the limit we can come 31 00:01:42,096 --> 00:01:45,022 up with a better answer for what it's approaching. 32 00:01:45,030 --> 00:01:47,022 Since we're just starting with these limit problems, 33 00:01:47,030 --> 00:01:48,067 let me draw a graph. 34 00:01:48,072 --> 00:01:51,000 I think this is gonna give you the intuition 35 00:01:51,007 --> 00:01:52,040 for what we're doing. 36 00:01:52,046 --> 00:01:54,039 And it'll probably give you the answer, 37 00:01:54,046 --> 00:01:57,012 but then I'll show you how to solve this analytically. 38 00:01:57,019 --> 00:02:03,019 So, if I draw a graph, 39 00:02:03,027 --> 00:02:06,022 so these are the axes. 40 00:02:06,030 --> 00:02:09,030 I'll do the graphical and the analytical 41 00:02:09,037 --> 00:02:10,093 at the same time. 42 00:02:11,000 --> 00:02:13,030 So, I'm gonna rewrite this expression in a way 43 00:02:13,037 --> 00:02:15,027 that maybe I can simplify it. 44 00:02:15,032 --> 00:02:17,012 So 2x + 2, 45 00:02:17,020 --> 00:02:23,080 isn't that the same thing as 2 times (x + 1)? 46 00:02:25,026 --> 00:02:28,002 2 times (x + 1), right? 47 00:02:28,009 --> 00:02:31,036 2x + 2 is the same thing as 2 times (x + 1), 48 00:02:31,043 --> 00:02:38,087 and then all of that is over, all of that is over x + 1, 49 00:02:38,092 --> 00:02:43,056 so as long as this expression and this expression 50 00:02:43,062 --> 00:02:47,079 don't equal 0, it actually turns out that this function, 51 00:02:47,086 --> 00:02:51,026 let's say that this is f(x), right? 52 00:02:51,032 --> 00:02:57,046 This function, well, for every value other than x = -1, 53 00:02:57,052 --> 00:02:59,076 you could actually cancel this and this out, 54 00:02:59,082 --> 00:03:05,009 and so really we see that f(x) is equal to-- 55 00:03:05,016 --> 00:03:07,043 I need to find a better tool-- 56 00:03:07,050 --> 00:03:12,070 f(x) = 2 57 00:03:12,076 --> 00:03:17,090 when x does not equal -1. 58 00:03:17,097 --> 00:03:22,007 And we saw when x is equal to -1, it's undefined; 59 00:03:22,013 --> 00:03:26,069 so undefined, undefined. 60 00:03:26,076 --> 00:03:31,092 Undefined when x = -1. 61 00:03:32,000 --> 00:03:33,067 So how would we graph that? 62 00:03:33,073 --> 00:03:35,063 We saw that f(x) = 2 63 00:03:35,069 --> 00:03:37,069 when x is not equal to -1, 64 00:03:37,076 --> 00:03:39,092 and f(x) is undefined when x = -1. 65 00:03:40,000 --> 00:03:42,060 And once again, all I did was kind of rewrite 66 00:03:42,067 --> 00:03:44,060 this exact same function, right? 67 00:03:44,067 --> 00:03:47,069 I showed that I could simplify and I could divide the numerator 68 00:03:47,076 --> 00:03:51,097 and denominator by x + 1 as long as x does not equal -1 69 00:03:52,002 --> 00:03:53,079 and then otherwise, it's undefined. 70 00:03:53,086 --> 00:03:55,019 So let me graph this. 71 00:03:55,026 --> 00:03:58,066 And I'll get a different color, maybe I'll go with red. 72 00:03:58,073 --> 00:04:05,010 So this is 2, so we see that x is-- and then this is -1. 73 00:04:05,016 --> 00:04:10,090 So for every other value other than -1, the value of this, 74 00:04:10,097 --> 00:04:13,013 of f(x) = 2. 75 00:04:13,020 --> 00:04:16,013 This is just, you know, this is 1, this is 2, 76 00:04:16,019 --> 00:04:21,036 this is 3--the pen stops, this is 3. 77 00:04:21,043 --> 00:04:26,019 So at -1 the graph is undefined, so there's a hole there, 78 00:04:26,026 --> 00:04:29,076 and that we keep going to the left-hand side. 79 00:04:29,083 --> 00:04:33,067 So if we're gonna do the limit, we can just visually say, 80 00:04:33,072 --> 00:04:37,039 well, as x--let me do another color now, 81 00:04:37,047 --> 00:04:39,080 as x comes from the left-hand side, 82 00:04:39,086 --> 00:04:43,012 as x goes from this side, 83 00:04:43,019 --> 00:04:44,067 what does f(x) equal? 84 00:04:44,072 --> 00:04:46,062 Well, f(x) is 2, 2, 2, 2, 2, 85 00:04:46,069 --> 00:04:51,036 f(x) = 2 until we get to exactly -1, right? 86 00:04:51,043 --> 00:04:53,062 And similarly, when we go from the other hand, 87 00:04:53,069 --> 00:04:55,007 it's the exact same thing. 88 00:04:55,012 --> 00:04:58,060 f(x) is 2, 2, 2, until we get to -1. 89 00:04:58,067 --> 00:05:02,019 So, you'll see, and make sure you see the visual here, 90 00:05:02,026 --> 00:05:08,019 that the limit as x approaches -1 91 00:05:08,026 --> 00:05:10,056 of (2 x + 2)/(x + 1), 92 00:05:10,062 --> 00:05:12,022 it equals 2. 93 00:05:12,029 --> 00:05:15,049 Let me draw a line here so you don't get mixed up. 94 00:05:15,057 --> 00:05:19,093 And I'm not formally, I guess, proving here 95 00:05:20,000 --> 00:05:21,043 that the limit is 2, 96 00:05:21,050 --> 00:05:23,023 but I'm showing you an analytical way, 97 00:05:23,029 --> 00:05:26,016 and this is how it tends to be done in algebra class, 98 00:05:26,023 --> 00:05:29,080 is that you tend to simplify the expression so you say, 99 00:05:29,086 --> 00:05:34,000 if there wasn't a hole here, what would the f(x) equal? 100 00:05:34,007 --> 00:05:36,023 And then, you just evaluate it at that point. 101 00:05:36,029 --> 00:05:38,056 I think this might give you a little intuition, 102 00:05:38,062 --> 00:05:42,012 but this isn't a formal solution, 103 00:05:42,019 --> 00:05:44,056 unless you're asked to, 104 00:05:44,062 --> 00:05:47,082 you tend not to be asked for a formal solution, 105 00:05:47,089 --> 00:05:49,049 you're asked what the limit is, 106 00:05:49,057 --> 00:05:51,083 and this is the way you can solve it. 107 00:05:51,089 --> 00:05:53,049 Actually another way that you could-- 108 00:05:53,057 --> 00:05:57,000 I often used to check my answers when I used to do it is 109 00:05:57,007 --> 00:05:59,060 you could take a calculator and try in, you know, 110 00:05:59,067 --> 00:06:07,083 what happens when-- what is f(-1.001), right? 111 00:06:08,067 --> 00:06:15,009 And you could also try what is f(-.999), right? 112 00:06:15,016 --> 00:06:17,033 'Cause what you wanna do is you wanna say, 113 00:06:17,040 --> 00:06:20,033 what is the function equal when I get really close to -1, 114 00:06:20,040 --> 00:06:23,047 and then you could keep going closer and closer to -1 115 00:06:23,052 --> 00:06:25,019 and see what the function approaches, 116 00:06:25,026 --> 00:06:27,076 and in this case you'll see that it approaches 2. 117 00:06:27,083 --> 00:06:32,057 So let's do another problem. 118 00:06:32,062 --> 00:06:38,057 Well let's say what is the limit? 119 00:06:38,062 --> 00:06:40,062 My pen, something wrong there, 120 00:06:40,069 --> 00:06:50,033 what is the limit as x approaches 0 of 1/x ? 121 00:06:51,060 --> 00:06:55,007 I think here it might be useful to draw this graph 122 00:06:55,012 --> 00:06:58,022 'cause it'll give you a visual representation. 123 00:06:58,029 --> 00:06:59,086 Actually, let's do it both ways. 124 00:06:59,093 --> 00:07:01,080 Let's do it the picking numbers method 125 00:07:01,086 --> 00:07:03,076 'cause I think that'll give you an intuition, 126 00:07:03,083 --> 00:07:05,093 and maybe it'll help us draw the graph. 127 00:07:06,000 --> 00:07:09,052 So, let's say that this is f(x), let's say that f(x), 128 00:07:09,060 --> 00:07:14,012 f(x)--you can tell my presentation is very unplanned, 129 00:07:14,019 --> 00:07:17,062 f(x) = 1/x. 130 00:07:17,069 --> 00:07:20,012 And we wanna find the limit as x approaches 0. 131 00:07:20,019 --> 00:07:26,016 So what is f of-- actually let's make a table. 132 00:07:26,083 --> 00:07:32,037 f(x). 133 00:07:36,019 --> 00:07:41,086 So clearly when x = 0 we don't know, it's undefined, 134 00:07:41,093 --> 00:07:45,043 1/0 is undefined. 135 00:07:45,050 --> 00:07:50,047 So we get undefined. 136 00:07:52,007 --> 00:07:58,033 Well, what happens when x = -.01? 137 00:07:59,007 --> 00:08:03,047 Well, when it's -.01, 1/-.01 138 00:08:03,052 --> 00:08:07,069 that is equal to -100, right? 139 00:08:07,076 --> 00:08:13,002 What happens when x = -.001, right? 140 00:08:13,009 --> 00:08:15,002 So we're getting closer and closer to 0 141 00:08:15,009 --> 00:08:16,046 from the negative direction. 142 00:08:16,052 --> 00:08:21,057 Well, here it equals-- gotta make sure my pen is working, 143 00:08:21,062 --> 00:08:26,050 color right, something's wrong with my tool. 144 00:08:28,047 --> 00:08:31,087 My computer is breaking down, I see what's going on. 145 00:08:38,003 --> 00:08:40,027 I think my computer just froze. 146 00:08:40,033 --> 00:08:43,057 Well, I'm gonna try to solve this and in the very next video, 147 00:08:43,062 --> 00:08:46,023 I'm gonna continue on with this problem. 148 00:08:46,029 --> 00:08:48,026 So, I'll see you at the next presentation 149 00:08:48,033 --> 00:08:51,082 while I figure out why my pen isn't working 150 00:08:51,090 --> 00:08:53,070 and then we'll continue with this problem. 151 00:08:53,076 --> 00:08:56,006 See you very soon.