1 00:00:00,436 --> 00:00:12,569 Let's say that f of x is equal to x squared plus x minus six over x minus two. 2 00:00:12,569 --> 00:00:21,509 And we're curious about what the limit of f of x as x approaches two is equal to. 3 00:00:21,509 --> 00:00:28,102 Now, the first attempt that you might want to do right when you see something like this is just see what happens, what is f of 2? 4 00:00:28,102 --> 00:00:34,797 Now, this won't always be the limit, even if it is defined, but it is a good place to start, just to see if something reasonable could pop out. 5 00:00:34,797 --> 00:00:42,548 So looking at it this way, if we just evaluate f of 2, in our numerator we're going to get two squared plus two minus six. 6 00:00:42,548 --> 00:00:49,236 So, that's going to be four plus two, which is six, minus six, so you're going to get zero in the numerator, and you're going to get zero in the denominator. 7 00:00:49,236 --> 00:00:58,236 So, we don't have - the function is not defined at x is equal to two. F not defined. 8 00:00:58,236 --> 00:01:07,236 So there's no simple thing there. Even if this did evaluate, if it was a continuous function, then actually the limit would be whatever the function is, but that isn't necessarily the case. 9 00:01:07,236 --> 00:01:10,462 But we see very clearly that the function is not defined here. 10 00:01:10,462 --> 00:01:14,706 So, let's see if we can simplify this, and we'll also try to graph it in some way. 11 00:01:14,706 --> 00:01:18,908 So one thing that might have jumped out at your head is you might want to factor this expression on top. 12 00:01:18,908 --> 00:01:24,502 So, if we want to rewrite this, we could rewrite the top expression - this just goes back to your algebra I - 13 00:01:24,502 --> 00:01:36,635 two numbers whose product is negative six and whose sum is positive three - well, that could be positive three and negative two, so this could be x plus three times x minus two. 14 00:01:36,635 --> 00:01:41,014 All of that over x minus two. 15 00:01:41,014 --> 00:01:58,234 So, as long as x does not equal two, these two things will cancel out, so we could say this is equal to x plus three for all x's except for x is equal to two - as long as x does not equal two. 16 00:01:58,234 --> 00:01:59,680 So that's another way of looking at it. 17 00:01:59,680 --> 00:02:14,414 Now, another way we could rewrite our f of x - blue, just to change the colors - we could rewrite f of x - this is the exact same function - f of x is equal to x plus three when x does not equal two, 18 00:02:14,414 --> 00:02:20,413 and we could even say it's undefined when x is equal to two. 19 00:02:20,413 --> 00:02:29,413 So given this definition, it becomes much clearer to us how we can actually graph f of x, so let's try to do it. 20 00:02:29,413 --> 00:02:35,012 So that is - that is not anywhere near being a straight line - that is much better. 21 00:02:35,012 --> 00:02:38,879 Let's call this the y axis - we'll call it y equals f of x. 22 00:02:38,879 --> 00:02:45,283 And then let's - over here let me make a horizontal line that is my x axis. 23 00:02:45,283 --> 00:02:49,152 So defined this way, f of x is equal to x plus three. 24 00:02:49,152 --> 00:02:57,758 So if this is one, two, three, we have a y-intercept at three and the slope is one. 25 00:02:57,758 --> 00:03:01,364 And it is defined for all x's except for x equal to two. 26 00:03:01,364 --> 00:03:04,156 So this is x equal to one, this is x equal to two. 27 00:03:04,156 --> 00:03:17,358 So when x is equal to two, it is undefined, so it is undefined right over there. 28 00:03:17,358 --> 00:03:23,230 So this is what f of x looks like. 29 00:03:23,230 --> 00:03:28,491 Now, given this, let's try to answer our question: what is the limit of f of x as x approaches two? 30 00:03:28,491 --> 00:03:38,246 Well, we could look at this graphically. As x approaches two from lower values than two, so this right over here is x is equal to two. 31 00:03:38,246 --> 00:03:43,695 If we get to maybe, let's say this is one point seven, we see that our f of x is right over there. 32 00:03:43,695 --> 00:03:52,163 If we get to one point nine, our f of x is right over there, so it seems to be approaching this value right over there. 33 00:03:52,163 --> 00:04:02,632 Similarly, as we approach two from values greater than it, if we're at like - I don't know - this could be, like, two point five, our f of x is right over there. 34 00:04:02,632 --> 00:04:10,426 If we get even closer to two, our f of x is right over there and once again we look like we are approaching this value. 35 00:04:10,426 --> 00:04:16,835 Or, another way of thinking about it, if we ride this line from the positive direction, we seem to be approaching this value for f of x. 36 00:04:16,835 --> 00:04:25,031 If we ride this line from the negative direction - from values less than two - we seem to be approaching this value right over here. 37 00:04:25,031 --> 00:04:31,292 And this is essentially the value of x plus three if we set x equal to two. 38 00:04:31,292 --> 00:04:45,759 This is essentially going to be - this value right over here is equal to five if we just look at it visually. If we just graphed a line with slope one with a y intercept of three, this value right over here is five. 39 00:04:45,759 --> 00:04:52,357 Now, we can also try to do this numerically, so let's try to do that. 40 00:04:52,357 --> 00:05:03,959 So, if this is our function definition, completely identical to our original definition, then we just try values as x gets closer and closer to two. 41 00:05:03,959 --> 00:05:13,959 So let's try values less than two. So, one point nine, nine, nine, nine - this is almost obvious - one point nine, nine, nine, nine plus three, 42 00:05:13,959 --> 00:05:16,497 well that gets you pretty darn close to five. 43 00:05:16,497 --> 00:05:20,425 If I put even more nine's here, and get even closer to two, we'd get even closer to five here. 44 00:05:20,425 --> 00:05:29,169 If we approach two from the positive direction, and then once again we're getting closer and closer to five from the positive direction. 45 00:05:29,169 --> 00:05:32,358 If we were even closer to two, we'd be even closer to five. 46 00:05:32,358 --> 00:05:36,358 So, whether we look at it numerically or we look at it graphically, it looks pretty clear that the limit here is going to be equal to five.