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Limit expression for the derivative of function (graphical)
With the graph of the function f as an aid, evaluate the following limits. So the first one is the limit as x approaches 3 of f of x minus f of 3 over x minus 3. So let's think about x minus-- x equals 3 is right over here. This right over here is f of 3, or we could say f of 3 is 1 right over here. That's the point 3 comma f of 3. And they're essentially trying to find the slope between an arbitrary x and that point as that x gets closer and closer to 3. So we can imagine an x that is above 3, that is, say, right over here. Well, if we're trying to find the slope between this x comma f of x and 3 comma f of 3, we see that it gets this exact same form. Your end point is f of x. So it's f of x minus f of 3 is your change in the vertical axis. That's this distance right over here. And we would divide by your change in the horizontal axis, which is your change in x. And that's going to be x minus 3. So that's the exact expression that we have up here when I picked this as an arbitrary x. And we see that that slope, just by looking at the line between those two intervals, seems to be negative 2. And the slope was the same thing if we go the other side. If x was less than 3, then we also would have a slope of negative 2. Either way, we have a slope of negative 2. And that's important because this limit is just the limit as x approaches 3. So it can be as x approaches 3 from the positive direction or from the negative direction. But in either case, the slope, as we get closer and closer to this point right over here, is negative 2. Now let's think about what they're asking us here. So we have 8, f of 8. So let's think. We have 8. This is 8 comma f of 8. So that's 8 comma f of 8 right over there. And they have f of 8 plus h. So our temptation might be to say, hey, 8 plus h is going to be someplace out here. It's going to be something larger than 8. But notice, they have the limit as h approaches 0 from the negative direction. So approaching 0 from the negative direction means you're coming to 0 from below. You're at negative 1, negative 0.5, negative 0.1, negative 0.0001. So h is actually going to be a negative number. So 8 plus h would actually be-- we could just pick an arbitrary point. It could be something like this right over here. So this might be the value of 8 plus h. And this would be the value of f of 8 plus h. So once again, they're finding-- or this expression is the slope between these two points. And then we are taking the limit as h approaches 0 from the negative direction. So as h gets closer and closer to 0, this down here moves further and further to the right. And these points move closer and closer and closer together. So this is really just an expression of the slope of the line, roughly-- and we see that it's constant. So what's the slope of the line over this interval? Well, you can just eyeball it and see, well, look. Every time x changes by 1, our f of x changes by 1. So the slope of the line there is 1. It would have been a completely different thing if this said limit as h approaches 0 from the positive direction. Then we would be looking at points over here. And we would see that we would slowly approach, essentially, a vertical slope, kind of an infinite slope.