Estimating limit numerically | Limits | Differential Calculus | Khan Academy

Consider the table with function values for ( f(x) = \frac{x^2}{1 - \cos 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(x) ) at ( x = 0.1 ) and enter this number in the table, rounded to the nearest thousandth.

From the table, what does the one-sided limit ( \lim_{x \to 0^+} f(x) ) appear to be? So, let's see what they did. They evaluated when ( x = 1 ); ( f(x) = 2.175 ). When ( x ) gets even a little bit closer to 0, and once again we're approaching 0 from values larger than zero, that's what this little superscript positive tells us. We're at 0.5 and we're at 2.042.

Then, when we get even closer to 0, at ( x = 0.2 ), ( f(x) = 2.007 ). So, 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 ( \frac{x^2}{1 - \cos x} ) when ( x = 0.1 ). The first thing I want to do is 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 ( \frac{0.1^2}{1 - \cos(0.1)} ) and this gets me 2.0016. Let's see, they want us to round to the nearest thousandth, so that would be 2.002. Type that in: 2.002.

It looks like the limit is approaching; the limit is approaching 2. It's not approaching 2.005; we just crossed 2.005 from 2.007 to 2.002. 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. Go into graph mode. Let me redefine my function here.

So, let's see, it's going to be ( \frac{x^2}{1 - \cos 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. Let me go to the range, and let's see: I care about approaching ( x ) from the positive direction.

As long as I see values around 0, I should be fine. Actually, I could actually zoom in a little bit more. So, I can make my minimum ( x ) value 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 seems like we’re approaching two, so the ( y ) max can be much smaller. Let’s see. Let me 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 approaching from the positive direction or from the negative direction, it looks like the value of the function approaches 2. But in this problem, we're only caring about as we have ( x ) values that are approaching zero from values larger than zero.

This is the limit; this is the one-sided limit that we care about, but the 2 shows up right over here as well.