1 00:00:00,000 --> 00:00:00,800 2 00:00:00,800 --> 00:00:03,330 Let's do some more limit examples. 3 00:00:03,330 --> 00:00:05,250 So let's get another problem. 4 00:00:05,250 --> 00:00:19,859 If I had the limit as x approaches 3 of, let's say, 5 00:00:19,859 --> 00:00:31,189 x squared minus 6x plus 9 over x squared minus 9. 6 00:00:31,190 --> 00:00:33,450 So the first thing I like to do whenever I see any of these 7 00:00:33,450 --> 00:00:35,150 limits problems is just substitute the number in and 8 00:00:35,149 --> 00:00:36,412 see if I get something that makes sense, and 9 00:00:36,412 --> 00:00:37,669 then we'd be done. 10 00:00:37,670 --> 00:00:39,170 Well, usually we'd be done. 11 00:00:39,170 --> 00:00:41,490 I don't want to make these sweeping statements. 12 00:00:41,490 --> 00:00:43,880 If the function is continuous, we'd be done. 13 00:00:43,880 --> 00:00:47,250 But if we put the 3 in the numerator, we get 3 squared, 14 00:00:47,250 --> 00:00:51,039 which is 9, minus 18 plus 9. 15 00:00:51,039 --> 00:00:52,659 So that equals 0. 16 00:00:52,659 --> 00:00:55,509 And the denominator also-- let's see, 3 squared minus 17 00:00:55,509 --> 00:00:56,399 9, that also equals 0. 18 00:00:56,399 --> 00:01:00,969 So we don't like having 0/0. 19 00:01:00,969 --> 00:01:03,839 My pen tool is malfunctioning again. 20 00:01:03,840 --> 00:01:07,290 So we don't like getting 0, 0, 0, so is there any way we can 21 00:01:07,290 --> 00:01:12,010 simplify this expression to maybe get it to an expression 22 00:01:12,010 --> 00:01:15,109 that, when we evaluate it at x equals 3, we actually get 23 00:01:15,109 --> 00:01:16,640 something that makes sense? 24 00:01:16,640 --> 00:01:19,180 Well, whenever I see two of these polynomials here, and 25 00:01:19,180 --> 00:01:22,220 they look, just by inspecting them, relatively easy to 26 00:01:22,219 --> 00:01:25,219 factor, I like to factor them out because maybe there's the 27 00:01:25,219 --> 00:01:27,250 same factor in the numerator and the denominator, and 28 00:01:27,250 --> 00:01:28,609 then we can simplify it. 29 00:01:28,609 --> 00:01:32,109 So let's say that this is the same thing as-- that looks 30 00:01:32,109 --> 00:01:39,689 like it's x plus 3-- no, no, no, x minus 3. 31 00:01:39,689 --> 00:01:42,450 This is x minus 3. 32 00:01:42,450 --> 00:01:44,280 It actually looks like it's x minus 3 squared, but we're 33 00:01:44,280 --> 00:01:46,840 just going to write x minus 3 times x minus 3, which is, of 34 00:01:46,840 --> 00:01:48,820 course, x minus 3 squared. 35 00:01:48,819 --> 00:01:51,939 And then in the denominator, you know how to factor these, 36 00:01:51,939 --> 00:02:01,310 this is x plus 3 times x minus 3, all right? 37 00:02:01,310 --> 00:02:03,990 So the limit as x approaches 3 of this expression is the same 38 00:02:03,989 --> 00:02:07,959 thing as the limit as x approaches 3 of 39 00:02:07,959 --> 00:02:08,759 this expression. 40 00:02:08,759 --> 00:02:12,659 And, of course, there's nothing we can do to change the fact 41 00:02:12,659 --> 00:02:16,150 that this function, or this expression, is undefined 42 00:02:16,150 --> 00:02:17,360 at x equals 3. 43 00:02:17,360 --> 00:02:19,500 But if we can simplify it, we can figure out 44 00:02:19,500 --> 00:02:21,319 what it approaches. 45 00:02:21,319 --> 00:02:26,009 Well, if we assume that x is any number but 3, we can cross 46 00:02:26,009 --> 00:02:28,329 out these two terms because then they wouldn't be 0, right? 47 00:02:28,330 --> 00:02:31,020 It only is 0 when x is equal to 3 because-- so in the numerator 48 00:02:31,020 --> 00:02:32,990 and the denominator, we can cross this out. 49 00:02:32,990 --> 00:02:36,250 And we can say-- and I'm not being very rigorous here, but 50 00:02:36,250 --> 00:02:38,610 this is kind of how it's taught, and I think you get the 51 00:02:38,610 --> 00:02:43,180 intuition-- that this is the same thing as the limit as x 52 00:02:43,180 --> 00:02:49,200 approaches 3 of x minus 3 over x plus 3. 53 00:02:49,199 --> 00:02:52,329 Now let's just try to stick the x in and see what we get. 54 00:02:52,330 --> 00:02:54,010 Well, in the numerator, we get 3 minus 3. 55 00:02:54,009 --> 00:02:56,030 We still get 0. 56 00:02:56,030 --> 00:02:58,189 But in the denominator here, we get 6, right? 57 00:02:58,189 --> 00:02:59,520 3 plus 3 is 6. 58 00:02:59,520 --> 00:03:00,770 So now we get a good number. 59 00:03:00,770 --> 00:03:04,680 0 or 6, well, that's a real number, so it's 0. 60 00:03:04,680 --> 00:03:05,710 0/6 is 0. 61 00:03:05,710 --> 00:03:07,170 So that was interesting. 62 00:03:07,169 --> 00:03:09,659 The first time we did it, we got the answer 0/0. 63 00:03:09,659 --> 00:03:12,759 And now we get the answer 0 by simplifying. 64 00:03:12,759 --> 00:03:15,590 But, of course, it's very important to remember that 65 00:03:15,590 --> 00:03:18,810 this expression is not defined at x equals 3. 66 00:03:18,810 --> 00:03:21,280 It's defined everywhere but, but if we were to graph it, and 67 00:03:21,280 --> 00:03:24,069 I encourage you to do so, you would see that as you get 68 00:03:24,069 --> 00:03:26,979 closer and closer to x equals 3, the value of this 69 00:03:26,979 --> 00:03:28,959 expression will equal 0. 70 00:03:28,960 --> 00:03:30,060 And I know what you're thinking. 71 00:03:30,060 --> 00:03:31,349 Well, this was 0/0. 72 00:03:31,349 --> 00:03:38,189 Is every time I get 0/0 going to end up just becoming 0 when 73 00:03:38,189 --> 00:03:39,300 I evaluate the expression? 74 00:03:39,300 --> 00:03:42,240 Well, let's explore that. 75 00:03:42,240 --> 00:03:45,570 Let me clear this. 76 00:03:45,569 --> 00:03:56,079 Let's say what is-- pen is not working-- the limit as x 77 00:03:56,080 --> 00:04:17,860 approaches 1 of x squared minus x minus 2. 78 00:04:17,860 --> 00:04:21,650 79 00:04:21,649 --> 00:04:23,639 No, let's say x squared plus x minus 2. 80 00:04:23,639 --> 00:04:26,110 As you can see, I do all this in my head, and 81 00:04:26,110 --> 00:04:27,449 I'm prone to mistakes. 82 00:04:27,449 --> 00:04:32,370 And all of that over x minus 1. 83 00:04:32,370 --> 00:04:33,939 Well, once again, if we just evaluate it, let's see what 84 00:04:33,939 --> 00:04:34,930 happens when x equals 1. 85 00:04:34,930 --> 00:04:38,199 You get 1 squared plus 1, so it's 2 minus 2. 86 00:04:38,199 --> 00:04:40,009 You get 0/0. 87 00:04:40,009 --> 00:04:44,639 So once again, we get 0/0, and we have to do something to 88 00:04:44,639 --> 00:04:46,959 this maybe to simplify it. 89 00:04:46,959 --> 00:04:48,620 Well, let's factor the top. 90 00:04:48,620 --> 00:04:54,530 So that's the same thing as the limit as x approaches 1. 91 00:04:54,529 --> 00:05:01,449 Well, that's x minus 1 times x plus 2, right? 92 00:05:01,449 --> 00:05:07,019 93 00:05:07,019 --> 00:05:09,839 And I think you'll often discover when you see a lot of 94 00:05:09,839 --> 00:05:13,719 limit problems that even if this top factor, if this top 95 00:05:13,720 --> 00:05:16,710 expression, is hard to factor, chances are, one of the things 96 00:05:16,709 --> 00:05:19,389 in the denominator that are making this expression 97 00:05:19,389 --> 00:05:21,769 undefined is probably a factor up here. 98 00:05:21,769 --> 00:05:24,389 So sometimes you might get a more complex thing that isn't 99 00:05:24,389 --> 00:05:27,509 as easy to factor as this, but a good starting point is to 100 00:05:27,509 --> 00:05:30,839 guess that one of the factors is going to be in the bottom 101 00:05:30,839 --> 00:05:33,359 expression because that's kind of the trick of these problems, 102 00:05:33,360 --> 00:05:35,750 to just simplify the expression. 103 00:05:35,750 --> 00:05:39,389 So once again, if we assume that x does not equal 1, and 104 00:05:39,389 --> 00:05:43,439 this expression would not be 0 and this would not be 0, 105 00:05:43,439 --> 00:05:47,230 then these two could be canceled out. 106 00:05:47,230 --> 00:05:51,009 And we get that this is just the same thing as the limit as 107 00:05:51,009 --> 00:05:55,069 x approaches 1 of x plus 2. 108 00:05:55,069 --> 00:05:56,029 Well, now this is pretty easy. 109 00:05:56,029 --> 00:05:58,849 What's the limit as x approaches 1 of x plus 2? 110 00:05:58,850 --> 00:06:03,180 Well, you just stick 1 in there, and you get 3. 111 00:06:03,180 --> 00:06:04,009 So it's interesting. 112 00:06:04,009 --> 00:06:07,839 When we just tried to evaluate the expression at 113 00:06:07,839 --> 00:06:10,579 x equals 1, we got 0/0. 114 00:06:10,579 --> 00:06:14,839 And in the previous example, we saw that it evaluated out when 115 00:06:14,839 --> 00:06:17,869 you simplified it to 0, and in this example, it came out to 3. 116 00:06:17,870 --> 00:06:19,759 And I really encourage you, if you have a graphing calculator, 117 00:06:19,759 --> 00:06:23,110 graph these functions that we're doing and see and show 118 00:06:23,110 --> 00:06:26,080 yourself visually that it's true, that the limit as you 119 00:06:26,079 --> 00:06:30,189 approach, say, x equals 1 actually does approach the 120 00:06:30,189 --> 00:06:31,949 limits that were solving for. 121 00:06:31,949 --> 00:06:34,599 And make up your own problems. 122 00:06:34,600 --> 00:06:36,939 Hell, that's what I'm doing. 123 00:06:36,939 --> 00:06:38,649 So you could prove it to yourself. 124 00:06:38,649 --> 00:06:40,319 So let's do another. 125 00:06:40,319 --> 00:06:42,279 Let's do one that I think is pretty interesting. 126 00:06:42,279 --> 00:06:48,019 127 00:06:48,019 --> 00:06:57,769 Let's say what's the limit as x approaches infinity? 128 00:06:57,769 --> 00:07:04,299 The limit as x approaches infinity of, let's say, x 129 00:07:04,300 --> 00:07:17,650 squared plus 3 over x to the third. 130 00:07:17,649 --> 00:07:19,959 So the way I think about these problems as they approach 131 00:07:19,959 --> 00:07:21,909 infinity, just think about what happens when you get 132 00:07:21,910 --> 00:07:24,960 really, really, really large values of x. 133 00:07:24,959 --> 00:07:28,029 And kind of a cheating way of doing this is, if you have a 134 00:07:28,029 --> 00:07:29,839 calculator, even if you don't have a calculator, put 135 00:07:29,839 --> 00:07:31,119 in huge numbers here. 136 00:07:31,120 --> 00:07:34,610 See what happens when x is a million, see what happens when 137 00:07:34,610 --> 00:07:36,900 x is a billion, see what happens when x is a trillion, 138 00:07:36,899 --> 00:07:38,029 and I think you'll get the point. 139 00:07:38,029 --> 00:07:40,269 You'll see what-- if there is a limit here, you'll 140 00:07:40,269 --> 00:07:41,500 see what it's going to. 141 00:07:41,500 --> 00:07:44,089 But the way I think about it is, in the numerator, kind of 142 00:07:44,089 --> 00:07:48,179 the fastest-growing term here is the x squared term, right? 143 00:07:48,180 --> 00:07:50,970 This is the fastest-growing term here. 144 00:07:50,970 --> 00:07:52,820 In the denominator, what's the fastest-growing term? 145 00:07:52,819 --> 00:07:54,474 Well, in the denominator, the fastest-growing term 146 00:07:54,475 --> 00:07:56,439 is this x to the third. 147 00:07:56,439 --> 00:07:58,160 Well, what's going to grow faster, x to the 148 00:07:58,160 --> 00:08:00,000 third or x squared? 149 00:08:00,000 --> 00:08:01,839 Well, yeah, x to the third's going to grow a lot 150 00:08:01,839 --> 00:08:03,129 faster than x squared. 151 00:08:03,129 --> 00:08:06,310 So this denominator, as you get larger and larger and larger 152 00:08:06,310 --> 00:08:10,230 values of x, is going to grow a lot faster than that numerator. 153 00:08:10,230 --> 00:08:12,700 So you could imagine if the denominator's growing much, 154 00:08:12,699 --> 00:08:14,904 much, much faster than the numerator, as you get larger 155 00:08:14,904 --> 00:08:17,129 and larger numbers, you're going to get a smaller and 156 00:08:17,129 --> 00:08:18,629 smaller and smaller fraction, right? 157 00:08:18,629 --> 00:08:20,279 It's going to approach 0. 158 00:08:20,279 --> 00:08:26,589 And so as you go to infinity, it approaches 0. 159 00:08:26,589 --> 00:08:29,529 I know that I kind of just hand waved, but that's really 160 00:08:29,529 --> 00:08:30,319 how you think about it. 161 00:08:30,319 --> 00:08:32,830 Another way you could do it is you could actually 162 00:08:32,830 --> 00:08:34,940 divide this fraction. 163 00:08:34,940 --> 00:08:37,470 You could actually divide this rational expression, and you'll 164 00:08:37,470 --> 00:08:39,750 get something like 1/x plus something, something, 165 00:08:39,750 --> 00:08:42,620 something, and then you'd also see, oh, well, the limit as x 166 00:08:42,620 --> 00:08:45,269 approaches infinity of 1/x is also 0. 167 00:08:45,269 --> 00:08:46,899 Let's do one more. 168 00:08:46,899 --> 00:08:49,189 I'll do this fast so I can confuse you. 169 00:08:49,190 --> 00:09:00,490 The limit as x approaches infinity of 3x squared plus 170 00:09:00,490 --> 00:09:05,669 x over 4x squared minus 5. 171 00:09:05,669 --> 00:09:08,169 172 00:09:08,169 --> 00:09:10,679 These problems kind of look confusing sometimes, but 173 00:09:10,679 --> 00:09:11,479 they're really easy. 174 00:09:11,480 --> 00:09:13,330 You just have to think about what happens as you get 175 00:09:13,330 --> 00:09:14,700 really large values of x. 176 00:09:14,700 --> 00:09:19,320 Well, as you get really large values of x, these small terms, 177 00:09:19,320 --> 00:09:21,670 these ones that don't grow as fast as these large terms, 178 00:09:21,669 --> 00:09:23,639 kind of don't matter anymore, right, because you're getting 179 00:09:23,639 --> 00:09:25,269 really large values of x. 180 00:09:25,269 --> 00:09:28,210 And this case, these don't matter anymore, and then 181 00:09:28,210 --> 00:09:32,210 these two x terms grow at the same pace, right? 182 00:09:32,210 --> 00:09:34,070 And they'll always be kind of growing in 183 00:09:34,070 --> 00:09:35,000 this ratio of 3 to 4. 184 00:09:35,000 --> 00:09:37,870 So the limit here is actually that easy. 185 00:09:37,870 --> 00:09:39,879 It's 3/4. 186 00:09:39,879 --> 00:09:41,320 So what you do is you just figure out what's the 187 00:09:41,320 --> 00:09:43,940 fastest-growing term on the top, what's the fastest-growing 188 00:09:43,940 --> 00:09:47,000 term on the bottom, and then figure out what it approaches. 189 00:09:47,000 --> 00:09:49,679 If they're the same term, then they kind of cancel out, and 190 00:09:49,679 --> 00:09:51,699 you say the limit approaches 3/4. 191 00:09:51,700 --> 00:09:54,410 It's a very nonrigorous way of doing it, but it gets 192 00:09:54,409 --> 00:09:55,569 you the right answer. 193 00:09:55,570 --> 00:09:57,370 See you in the next presentation. 194 00:09:57,370 --> 00:09:57,500