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One-sided limits from tables

Video transcript

- [Instructor] Consider the table with function values for f of x is equal to x squared over one minus cosine x at positive x-values near zero. Notice that there is one missing value in the table. This is the missing one right here. Use a calculator to evaluate f of x at x equals 0.1 and enter this number in the table rounded to the nearest thousandth. From the table, what does the one-sided limit, the limit as x approaches zero from the positive direction of x squared over one minus cosine of x appear to be? So let's see what they did. They evaluated when x equals one, f of x equals 2.175. When x gets even a little bit closer to zero, and once again, we're approaching zero from values larger than zero. That's what this little superscript positive tell us. We're at 0.5, and we're at 2.042. Then when we get even closer to zero, 0.2, f of x is 2.007. And so I'm guessing when I'm getting even closer, it's gonna be even closer to two right over here. But let's verify that. Get my calculator out. So I wanna evaluate x squared over one minus cosine of x when x is equal to 0.1. So the first thing I wanna actually, let me verify that I'm in radian mode 'cause otherwise I might get a strange answer. So I am in radian mode. And so let me evaluate it. So I'm going to have 0.1 squared over, let me do divide it by one minus cosine of 0.1. And this gets me 2.0016. And let's see, they want us to round to the nearest thousandth. So that'll be 2.002. Type that in. 2.002. And so it looks like the limit is approaching two. It's not approaching 2.005. We just crossed 2.005 from 2.007 to 2.002. So let's check our answer. And we got it right. And I always find it fun to visualize these things, and that's what a graphing calculator is good for. It can actually graph things. So let's graph this right over here. So go into graph mode. Let me redefine my function here. So let's see. It's going to be x squared divided by one minus cosine of x. And then let me make sure that the range of my graph is right, so I'm zoomed in at the right part that I care about. So let me go to the range. And let's see. I care about approaching zero from the positive direction, but as long as I see values around zero, I should be fine. But I could actually zoom in a little bit more, so I can make them my minimum x-value. I don't know, let me make it negative one. Let me make my maximum x-value. The maximum x-value here is one, but just to get some space here I'll make this 1.5. So the x-scale is one. Y-minimum, it seems like we're approaching two. So the y-max can be much smaller, let's see. We'll make y-max three. And now let's graph this thing. So let's see what it's doing. It looks like... And you see here whether, and actually it looks like whether you're approaching from the positive direction or from the negative direction, it looks like you're approaching, the value of the function approaches two. But this problem, we're only caring about as we have x-values that approaching zero from values larger than zero. So this is the limit. This is the one-sided limit that we care about with the two shows up right over here as well.