Limit expression for the derivative of cos(x) at a minimum point

Let g of x equal cosine of x. So let's actually graph that. So that would be the graph right over there. I drew it ahead of time. So this is a segment of cosine of x between x is equal to 0 and x equals pi. Obviously, it keeps on going in both directions after that. And then they ask us, what is the limit as h approaches 0 of g of pi plus h minus g of pi over h? So let's just think about it a little bit. So they are concerning themselves with g of pi. So let's look at the point pi comma g of pi. So that right there is the point pi comma g of pi. And they're also concerning themselves with g of pi plus h. So let's say that this right over here is the x value pi plus h. And then this right over here would be the point pi plus h, g of pi plus h-- would be this point right over here. And they're essentially trying to find the slope between these two points. If we wanted to find the slope between these points, it would be change in our y value over change in x, or change in the vertical over change in the horizontal. What is our change in the vertical? What is the change in the vertical? Well, the change in the vertical, we'll take this y value here-- so g of pi plus h. We'll subtract this y value here-- minus g of pi. So that was our change in the vertical over the change in the horizontal. Well, what is that going to be? Well that's going to be pi plus h minus pi. And this is exactly what we have or, before we take the limit, what we have right over here. So this is going to be g of pi plus h minus g of pi, all of that over-- these pi's cancel out-- all of that over h. Now, so this is the slope, this slope the way I've-- it's a slope of the secant line right over here. Now let's think about what is going to happen as h gets closer and closer to 0. So the way I've drawn it right over here, that means that this point is going to go further and further to the left. This point is going to go further and further and further to the left. So pi plus h, as h approaches 0, is going to approach pi. If we assumed h was a negative value, then we would be approaching from here. But what is going to happen as h becomes smaller and smaller and smaller? Well, the point pi plus h, g of pi plus h is going to get closer and closer to this point. And the slope of those secant lines-- I know it's hard to see, because it's small here-- they're going to start getting closer and closer to the slope of the tangent line right at x equals pi. So this right over here really is another way of saying the slope of the tangent line at x equals pi. Well, what does that look like? Well the slope of the tangent line, we're at a minimum point at x equals pi right over here. Cosine of pi we know is a negative 1. That's a minimum value. It's one of the minimum values. It keeps going to that for cosine of x. And so its tangent line is just going to be a horizontal line. So we know that this right over here is going to be 0. Now, there are other ways you could have tackled it. We don't have the tools right now to do it purely algebraically-- to say cosine of pi plus h minus cosine of pi. There are ways to do it. But we're not going to do that now. The other option that you could do is use a calculator. So for example, you could say, well, OK, let's just take very small h's and just evaluate them. So we'll evaluate cosine of pi plus some h minus cosine of pi over that small h. And we're going to get smaller and smaller h's here. And so actually, let's try to do that. That could be interesting. So let me clear this. So let's just take a really, really small h's, h's getting really, really close to 0. So let me make sure I'm in Radian mode, first of all. I'm in Degree. So let me fix that. All right, perfect. And now let me take cosine of pi. I'll take a reasonably small h, 0.1, minus cosine of pi. I always forget where the pi is. So that's the numerator. And then let me divide that by the same h, by 0.1. So this is just my previous answer divided by 0.1. So I get 0.04. Now let me make h even smaller. And I'll actually do in one expression, actually, this time. So cosine of-- actually, make it a lot smaller-- of pi plus 0.0001-- so 1/10,000 more than pi, right over there-- minus cosine of pi. And now we are going to divide by this h, 0.0001. And what do we get? 5 times 10 to the negative 5th-- so you see clearly that we're getting to some really, really, really, really small numbers, that this expression is approaching 0.