1 00:00:00,518 --> 00:00:02,439 Let's do a few more examples of finding 2 00:00:02,439 --> 00:00:04,369 the limit of functions as x approaches 3 00:00:04,369 --> 00:00:06,670 infinity or negative infinity. 4 00:00:06,670 --> 00:00:08,770 So here I have this crazy function, 5 00:00:08,770 --> 00:00:16,861 [reads function] 6 00:00:16,861 --> 00:00:19,260 So what's going to happen as x 7 00:00:19,260 --> 00:00:20,861 approaches infinity? 8 00:00:20,861 --> 00:00:22,011 And the key here, like we've seen 9 00:00:22,011 --> 00:00:24,091 in other examples, is just to realize 10 00:00:24,091 --> 00:00:26,421 which terms will dominate. 11 00:00:26,421 --> 00:00:28,041 So for example, in the numerator, 12 00:00:28,041 --> 00:00:29,372 out of these three terms, 13 00:00:29,372 --> 00:00:31,983 the 9x^7 is going to grow much faster 14 00:00:31,983 --> 00:00:34,442 than any of these other terms, 15 00:00:34,442 --> 00:00:37,643 so this is the dominating term 16 00:00:37,643 --> 00:00:40,014 in the numerator, and in the denominator, 17 00:00:40,014 --> 00:00:43,277 3x^7 is going to grow much faster than an x^5 term, 18 00:00:43,354 --> 00:00:44,771 and definitely much faster 19 00:00:44,817 --> 00:00:47,174 than a log base 2 term. 20 00:00:47,174 --> 00:00:49,914 So at infinity, as we get closer and closer to infinity, 21 00:00:49,914 --> 00:00:53,388 this function is going to be roughly equal to 22 00:00:53,449 --> 00:00:58,927 9x^7 over 3x^7, and so we can say, 23 00:00:59,082 --> 00:01:01,199 especially since as we get larger and larger, 24 00:01:01,292 --> 00:01:03,220 as we get closer and closer to infinity, 25 00:01:03,266 --> 00:01:04,868 these two things are going to get 26 00:01:04,915 --> 00:01:06,346 closer and closer to each other, 27 00:01:06,376 --> 00:01:07,851 we can say this limit is going to be 28 00:01:07,851 --> 00:01:10,660 the same thing as this limit, 29 00:01:10,660 --> 00:01:12,410 which is going to be equal to 30 00:01:12,410 --> 00:01:15,372 the limit as x approaches infinity - 31 00:01:15,372 --> 00:01:17,501 well, we can just cancel out the x^7's, 32 00:01:17,501 --> 00:01:20,441 so it's going to be 9/3, or just 3, 33 00:01:20,441 --> 00:01:22,251 which is just going to be 3. 34 00:01:22,251 --> 00:01:24,913 So that is our limit as x approaches infinity 35 00:01:24,913 --> 00:01:26,911 of all of this craziness. 36 00:01:26,911 --> 00:01:28,332 Now let's do the same with this function 37 00:01:28,332 --> 00:01:30,253 over here. Once again crazy function, 38 00:01:30,253 --> 00:01:31,503 we're going to negative infinity, 39 00:01:31,503 --> 00:01:33,083 but the same principles apply. 40 00:01:33,083 --> 00:01:36,413 Which terms dominate as the absolute value 41 00:01:36,413 --> 00:01:37,944 of x gets larger and larger and larger, 42 00:01:37,994 --> 00:01:40,117 as x gets larger in magnitude. 43 00:01:40,836 --> 00:01:40,837 Well in the numerator, it's the 3x^3 term, 44 00:01:43,443 --> 00:01:43,443 in the denominator it's the 6x^4 term, 45 00:01:49,630 --> 00:01:49,630 so this is going to be the same thing as 46 00:01:49,630 --> 00:01:49,631 the limit of 3x^3 over 6x^4 47 00:01:49,631 --> 00:01:55,665 as x approaches negative infinity. 48 00:01:55,665 --> 00:01:58,374 And if we simplify this, this is going to be equal to 49 00:01:58,374 --> 00:02:01,504 the limit as x approaches negative infinity 50 00:02:01,504 --> 00:02:05,507 of 1 over 2x. 51 00:02:05,507 --> 00:02:07,506 And what's this going to be? 52 00:02:07,506 --> 00:02:09,917 Well if the denominator, even though it's 53 00:02:09,917 --> 00:02:12,376 becoming a larger and larger negative number, 54 00:02:12,376 --> 00:02:13,777 it becomes one over 55 00:02:13,777 --> 00:02:16,417 a very, very large negative number, 56 00:02:16,417 --> 00:02:18,057 which is going to get us pretty darn close 57 00:02:18,057 --> 00:02:20,437 to zero, just as one over x as 58 00:02:20,437 --> 00:02:23,439 x approaches negative infinity gets us close to zero. 59 00:02:23,439 --> 00:02:24,908 So this right over here, 60 00:02:24,908 --> 00:02:26,307 the horizontal asymptote in this case 61 00:02:26,307 --> 00:02:28,857 is y is equal to 0, 62 00:02:28,857 --> 00:02:29,998 and I encourage you to graph it, 63 00:02:29,998 --> 00:02:32,709 or try it out with numbers to verify that for yourself. 64 00:02:32,709 --> 00:02:34,768 The key realization here is to 65 00:02:34,768 --> 00:02:37,079 simplify the problem by just thinking 66 00:02:37,079 --> 00:02:38,439 about which terms are going to 67 00:02:38,439 --> 00:02:42,109 dominate the rest. 68 00:02:42,109 --> 00:02:43,450 Now let's think about this one. 69 00:02:43,450 --> 00:02:45,171 What is the limit of this crazy function 70 00:02:45,171 --> 00:02:47,501 as x approaches infinity? 71 00:02:47,501 --> 00:02:49,851 Well once again, what are the dominating terms? 72 00:02:49,851 --> 00:02:51,250 In the numerator it's 4x^4, 73 00:02:51,250 --> 00:02:54,501 in the denominator it's 250x^3, 74 00:02:54,501 --> 00:02:56,171 these are the highest degree terms. 75 00:02:56,171 --> 00:02:57,850 So this is going to be the same thing 76 00:02:57,850 --> 00:03:00,300 as the limit as x approaches infinity 77 00:03:00,300 --> 00:03:09,052 of 4x^4 over 250x^3, 78 00:03:09,052 --> 00:03:11,252 which is going to be the same thing as 79 00:03:11,252 --> 00:03:13,002 the limit of - let's see - 80 00:03:13,002 --> 00:03:15,102 4 - well I could just - this is 81 00:03:15,102 --> 00:03:16,912 going to be the same thing as - 82 00:03:16,912 --> 00:03:18,504 well we could just divide 250 - well 83 00:03:18,504 --> 00:03:20,104 I'll just leave it like this. 84 00:03:20,104 --> 00:03:23,054 It's going to be the limit of 4 over 250 85 00:03:23,054 --> 00:03:24,974 - x^4 divided by x^3 is just x - 86 00:03:24,974 --> 00:03:26,913 times x, as x approaches infinity. 87 00:03:26,913 --> 00:03:31,914 Or, we could even say this is going to be 4/250 88 00:03:31,914 --> 00:03:40,175 times the limit as x approaches infinity of x. 89 00:03:40,175 --> 00:03:41,306 Now what's this? 90 00:03:41,306 --> 00:03:43,335 What's the limit of x as x approaches infinity? 91 00:03:43,335 --> 00:03:45,705 Well it's just going to keep growing forever, 92 00:03:45,705 --> 00:03:46,975 so this is just going to be - 93 00:03:46,975 --> 00:03:47,915 this right over here is just going to be 94 00:03:47,915 --> 00:03:49,915 infinity, and infinity times some number 95 00:03:49,915 --> 00:03:51,577 right over here is going to be infinity, 96 00:03:51,577 --> 00:03:53,778 so the limit as x approaches infinity 97 00:03:53,778 --> 00:03:55,336 of all of this is actually unbounded. 98 00:03:55,336 --> 00:03:57,707 It's infinity. 99 00:03:57,707 --> 00:03:58,977 And a kind of obvious way 100 00:03:58,977 --> 00:04:00,376 of seeing that right from the get-go 101 00:04:00,376 --> 00:04:02,768 is to realize that the numerator 102 00:04:02,768 --> 00:04:04,768 has a fourth degree term, 103 00:04:04,768 --> 00:04:06,419 while the highest degree term in the denominator 104 00:04:06,419 --> 00:04:07,918 is only a third degree term. 105 00:04:07,918 --> 00:04:09,437 So the numerator is going to grow 106 00:04:09,437 --> 00:04:11,338 far faster than the denominator. 107 00:04:11,338 --> 00:04:13,368 So if the numerator is growing far faster 108 00:04:13,368 --> 00:04:15,669 than the denomiator, 109 00:04:15,669 --> 00:04:17,979 you're going to approach infinity in this case. 110 00:04:17,979 --> 00:04:22,236 If the numerator is growing far slower 111 00:04:22,236 --> 00:04:23,946 than the denominator, 112 00:04:23,946 --> 00:04:25,617 if the denominator is growing far faster 113 00:04:25,617 --> 00:04:27,237 than the numerator, like this case, 114 00:04:27,237 --> 00:04:29,837 you are then approaching zero. 115 00:04:29,837 --> 99:59:59,999 So hopefully you find that a little bit useful.