1 00:00:00,000 --> 00:00:00,640 2 00:00:00,640 --> 00:00:08,810 We need to evaluate the limit, as x approaches infinity, of 4x 3 00:00:08,810 --> 00:00:18,350 squared minus 5x, all of that over 1 minus 3x squared. 4 00:00:18,350 --> 00:00:20,120 So infinity is kind of a strange number. 5 00:00:20,120 --> 00:00:22,940 You can't just plug in infinity and see what happens. 6 00:00:22,940 --> 00:00:25,450 But if you wanted to evaluate this limit, what you might try 7 00:00:25,449 --> 00:00:28,419 to do is just evaluate-- if you want to find the limit as this 8 00:00:28,420 --> 00:00:31,020 numerator approaches infinity, you put in really large numbers 9 00:00:31,019 --> 00:00:32,839 there, and you're going to see that it approaches infinity. 10 00:00:32,840 --> 00:00:34,980 That the numerator approaches infinity as 11 00:00:34,979 --> 00:00:36,199 x approaches infinity. 12 00:00:36,200 --> 00:00:38,300 And if you put really large numbers in the denominator, 13 00:00:38,299 --> 00:00:40,599 you're going to see that that also-- well, 14 00:00:40,600 --> 00:00:41,780 not quite infinity. 15 00:00:41,780 --> 00:00:45,230 3x squared will approach infinity, but we're 16 00:00:45,229 --> 00:00:46,189 subtracting it. 17 00:00:46,189 --> 00:00:49,099 18 00:00:49,100 --> 00:00:52,280 If you subtract infinity from some non-infinite number, it's 19 00:00:52,280 --> 00:00:54,219 going to be negative infinity. 20 00:00:54,219 --> 00:00:57,609 So if you were to just kind of evaluate it at infinity, 21 00:00:57,609 --> 00:01:00,259 the numerator, you would get positive infinity. 22 00:01:00,259 --> 00:01:04,914 The denominator, you would get negative infinity. 23 00:01:04,915 --> 00:01:06,260 So I'll write it like this. 24 00:01:06,260 --> 00:01:07,420 Negative infinity. 25 00:01:07,420 --> 00:01:09,799 And that's one of the indeterminate forms 26 00:01:09,799 --> 00:01:11,769 that L'Hopital's Rule can be applied to. 27 00:01:11,769 --> 00:01:13,039 And you're probably saying, hey, Sal, why are we even 28 00:01:13,040 --> 00:01:14,400 using L'Hopital's Rule? 29 00:01:14,400 --> 00:01:16,450 I know how to do this without L'Hopital's Rule. 30 00:01:16,450 --> 00:01:19,130 And you probably do, or you should. 31 00:01:19,129 --> 00:01:20,159 And we'll do that in a second. 32 00:01:20,159 --> 00:01:22,219 But I just wanted to show you that L'Hopital's Rule also 33 00:01:22,219 --> 00:01:24,150 works for this type of problem, and I really just wanted to 34 00:01:24,150 --> 00:01:28,190 show you an example that had a infinity over negative 35 00:01:28,189 --> 00:01:30,509 or positive infinity indeterminate form. 36 00:01:30,510 --> 00:01:32,320 But let's apply L'Hopital's Rule here. 37 00:01:32,319 --> 00:01:37,359 So if this limit exists, or if the limit of their derivatives 38 00:01:37,359 --> 00:01:43,379 exist, then this limit's going to be equal to the limit as x 39 00:01:43,379 --> 00:01:47,000 approaches infinity of the derivative of the numerator. 40 00:01:47,000 --> 00:01:49,750 So the derivative of the numerator is-- the derivative 41 00:01:49,750 --> 00:01:56,129 of 4x squared is 8x minus 5 over-- the derivative of the 42 00:01:56,129 --> 00:01:58,420 denominator is, well, derivative of 1 is 0. 43 00:01:58,420 --> 00:02:03,200 Derivative of negative 3x squared is negative 6x. 44 00:02:03,200 --> 00:02:06,200 And once again, when you evaluated infinity, the 45 00:02:06,200 --> 00:02:08,930 numerator is going to approach infinity. 46 00:02:08,930 --> 00:02:11,310 And the denominator is approaching negative infinity. 47 00:02:11,310 --> 00:02:14,240 Negative 6 times infinity is negative infinity. 48 00:02:14,240 --> 00:02:16,390 So this is negative infinity. 49 00:02:16,389 --> 00:02:18,239 So let's apply L'Hopital's Rule again. 50 00:02:18,240 --> 00:02:22,250 So if the limit of these guys' derivatives exist-- or the 51 00:02:22,250 --> 00:02:25,620 rational function of the derivative of this guy divided 52 00:02:25,620 --> 00:02:28,870 by the derivative of that guy-- if that exists, then this 53 00:02:28,870 --> 00:02:33,009 limit's going to be equal to the limit as x approaches 54 00:02:33,009 --> 00:02:38,280 infinity of-- arbitrarily switch colors-- derivative 55 00:02:38,280 --> 00:02:41,569 of 8x minus 5 is just 8. 56 00:02:41,569 --> 00:02:45,709 Derivative of negative 6x is negative 6. 57 00:02:45,710 --> 00:02:48,040 And this is just going to be-- this is just a constant here. 58 00:02:48,039 --> 00:02:49,909 So it doesn't matter what limit you're approaching, this is 59 00:02:49,909 --> 00:02:51,199 just going to equal this value. 60 00:02:51,199 --> 00:02:52,669 Which is what? 61 00:02:52,669 --> 00:02:55,539 If we put it in lowest common form, or simplified 62 00:02:55,539 --> 00:02:57,669 form, it's negative 4/3. 63 00:02:57,669 --> 00:03:01,809 64 00:03:01,810 --> 00:03:03,439 So this limit exists. 65 00:03:03,439 --> 00:03:04,919 This was an indeterminate form. 66 00:03:04,919 --> 00:03:11,059 And the limit of this function's derivative over this 67 00:03:11,060 --> 00:03:13,680 function's derivative exists, so this limit must also 68 00:03:13,680 --> 00:03:15,189 equal negative 4/3. 69 00:03:15,189 --> 00:03:17,609 And by that same argument, that limit also must be 70 00:03:17,610 --> 00:03:20,520 equal to negative 4/3. 71 00:03:20,520 --> 00:03:21,950 And for those of you who say, hey, we already 72 00:03:21,949 --> 00:03:23,539 knew how to do this. 73 00:03:23,539 --> 00:03:25,469 We could just factor out an x squared. 74 00:03:25,469 --> 00:03:26,400 You are absolutely right. 75 00:03:26,400 --> 00:03:28,439 And I'll show you that right here. 76 00:03:28,439 --> 00:03:30,579 Just to show you that it's not the only-- you know, 77 00:03:30,580 --> 00:03:32,920 L'Hopital's Rule is not the only game in town. 78 00:03:32,919 --> 00:03:35,629 And frankly, for this type of problem, my first reaction 79 00:03:35,629 --> 00:03:38,379 probably wouldn't have been to use L'Hopital's Rule first. 80 00:03:38,379 --> 00:03:42,359 You could have said that that first limit-- so the limit as x 81 00:03:42,360 --> 00:03:49,630 approaches infinity of 4x squared minus 5x over 1 minus 82 00:03:49,629 --> 00:03:55,829 3x squared is equal to the limit as x approaches infinity. 83 00:03:55,830 --> 00:03:59,940 Let me draw a little line here, to show you that this is equal 84 00:03:59,939 --> 00:04:02,009 to that, not to this thing over here. 85 00:04:02,009 --> 00:04:03,789 This is equal to the limit as x approaches infinity. 86 00:04:03,789 --> 00:04:06,019 Let's factor out an x squared out of the numerator 87 00:04:06,020 --> 00:04:06,689 and the denominator. 88 00:04:06,689 --> 00:04:15,750 So you have an x squared times 4 minus 5 over x. 89 00:04:15,750 --> 00:04:19,220 Right? x squared times 5 over x is going to be 5x. 90 00:04:19,220 --> 00:04:22,440 Divided by-- let's factor out an x out of the numerator. 91 00:04:22,439 --> 00:04:29,750 So x squared times 1 over x squared minus 3. 92 00:04:29,750 --> 00:04:33,540 And then these x squareds cancel out. 93 00:04:33,540 --> 00:04:37,920 So this is going to be equal to the limit as x approaches 94 00:04:37,920 --> 00:04:45,990 infinity of 4 minus 5 over x over 1 over x squared minus 3. 95 00:04:45,990 --> 00:04:47,759 And what's this going to be equal to? 96 00:04:47,759 --> 00:04:50,610 Well, as x approaches infinity-- 5 divided by 97 00:04:50,610 --> 00:04:52,050 infinity-- this term is going to be 0. 98 00:04:52,050 --> 00:04:54,160 Super duper infinitely large denominator, 99 00:04:54,160 --> 00:04:56,650 this is going to be 0. 100 00:04:56,649 --> 00:04:58,799 That is going to approach 0. 101 00:04:58,800 --> 00:04:59,900 And same argument. 102 00:04:59,899 --> 00:05:02,399 This right here is going to approach 0. 103 00:05:02,399 --> 00:05:05,009 All you're left with is a 4 and a negative 3. 104 00:05:05,009 --> 00:05:08,129 So this is going to be equal to negative 4 over a 105 00:05:08,129 --> 00:05:10,610 negative 3, or negative 4/3. 106 00:05:10,610 --> 00:05:13,270 So you didn't have to do use L'Hopital's Rule 107 00:05:13,269 --> 00:05:14,909 for this problem. 108 00:05:14,910 --> 00:05:15,000