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One-sided limits from tables

Video transcript
Consider the table with function values for f of x is equal to x squared over 1 minus cosine x at positive x-values near 0. 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 0 from the positive direction of x squared over 1 minus cosine of x, appear to be? So let's see what they did. They evaluated when x equals 1, f of x is 2.175. When x gets even a little bit closer to 0-- and once again we're approaching 0 from values larger than 0, that's what this little superscript positive tells us-- we're at 0.5 and we're at 2.042. Then when we even closer to 0, 0.2 f of x is 2.007. I'm guessing when I'm getting even closer it's going to be even closer to 2 right over here. But let's verify that. Get my calculator out. So I want to evaluate x squared over 1 minus cosine of x when x is equal to 0.1. Let me actually verify that I'm in radian mode, because otherwise, I might get a strange answer. So I am in radian mode. Let me evaluate it. So I'm going to have 0.1 squared divided by 1 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 would be 2.002. Type that in, 2.002. And so it looks like the limit is approaching 2. 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. 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 in to graph mode. Let me redefine my function here. So let's see, it's going to be x squared divided by 1 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 the part that I care about. So let me go to the range. And let's see, I care about approaching x from the-- or approaching 0 from the positive direction, but as long as I see values around 0, I should be fine. But I could actually zoom in a little bit more. So I could make my minimum x-value negative 1. Let me make my maximum x-value-- the maximum x-value here is 1, but just to get some space here I'll make this 1.5. So the x-scale is 1. y minimum, it seems like we're approaching 2. So the y max can be much smaller. Let's see, let me make y max 3. And now let's graph this thing. So let's see what it's doing. And actually it looks -- whether you're approaching from the positive direction or from the negative direction-- it looks like the value of the function approaches 2. But this problem, we're only caring about-- as we have x-values that are approaching 0 from values larger than 0. So this is the one-sided limit that we care about. But the 2 shows up right over here, as well.