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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 let's just think about it a little bit so they are concerning themselves with G of Pi so 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 G of PI plus h so let's say that this right over here is the x-value pi plus h 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 were wanted to find the slope between these points so 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 will take this this Y value here so G of PI plus H will subtract this Y value here minus G of Pi so that was our change in the vertical over the change in the horizontal over the change in the horizontal what is that going to be well that's going to be PI plus h PI plus h minus PI 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 this slope this slope the way I've it's the 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 zero 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 + H as H approaches zero is going to approach pi if H wasn't if we assumed H was a negative value that we would be approaching from here but what is going to happen as as h becomes smaller and smaller and smaller well G of pi plus h is going to is going to or 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 the slope of the secant line that was hard to seek is so small here is going to is going to become or 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 this right over here is another way of saying as the slope of the tangent line slope of tangent line at 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 negative one that's a minimum value it's one of the minimum values it keeps going to that for cosine of X and so it's tangent line is just going to be a horizontal line it's 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 you know 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 okay let's 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 we're going to get smaller and smaller each is here and so actually let's let's try to do that that could be interesting so let me clear this so let's just take really really small ages the H is getting really really close to zero so let me make sure I'm in Radian mode folks first of all I'm in degree so let me fix that all right perfect and now let me take cosine cosine of pi see a PI I'll take a reasonably small H point one minus cosine of PI cosine of PI cosine I always figure what 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 two point zero four now let me make H even smaller and I'll actually do it in one expression actually this time so cosine cosine of actually make it a lot smaller of Pi plus point zero zero zero one so one ten-thousandth more than pi right over there minus cosine of Phi minus cosine of PI and now we are going to divide by this H point zero zero zero one and what do we get five times 10 to the negative fifth so you see clearly that we're getting two really really really real all numbers that are this this expression is approaching zero