![]() We knew we could make things exact if we could bring the two points infinitely close to each other. (4) Now we wanted to know if we could make things exact. ![]() (Understanding why we got a better and better approximation was quite hard conceptual work.) Similarly students began to recognize graphically that the slope of two points really close to each other is actually almost the slope of the tangent line to the function. ![]() And the closer that we got, the better our approximation was. (3) Students came to understand that we could approximate the instantaneous rate of change by taking the slope of two points really really really close to each other on a function. What might an instantaneous rate of change mean? Is it an oxymoron to have a rate of change at a instant? So we have something that feels philosophically impossible but in our guts and everyday experience feels right. But we also recognized that we understand that instantaneous rates of change do exist, because we believe our speedometers in our car which say 60mph. So we problematized the idea of instantaneous rate of change. They saw it was problematic, because how can something be changing at an instant? If you say you’re travelling “58 mph at 2:03pm,” what exactly does that mean? There is no time interval for this 58mph to pop out of, since we’re talking about an instant, a single moment in time (of 2:03pm). (2) Students talked about the idea of instantaneous rate of change. (1) Students talked about average rate of change. All I really need - at least for derivatives - is how to find the limit as one single variable goes to 0. And this way I’m not wasting a whole quarter (or even half a quarter) with such a simple idea. Do I really need them to understand limits at infinity of rational functions? Or limits of piecewise functions? Or limits of things like as ? I figure if that’s all I need limits for, I can target how I introduce and use limits to really focus on those things. when you say you are taking the sum of an infinite sum of infinitely thin rectanglesĪnd… that’s pretty much it.when you use the formal definition of the derivativeĪnd… that’s pretty much it.But where do they show up in derivatives? Reasoning Behind My Decision to Eliminate Limitsįor me, calculus has two major parts: the idea of the derivative, and the idea of the integral. I’ll explain.įirst I’ll explain my reasoning behind this decision. And this year, I’ve reduced the time I spend on limits to about 5 minutes.* In more recent years, I spent maybe a sixth of a year on them. I used to spend a quarter of a year on them. Each year, they lose more and more value in my mind. In the past few years, I’ve done a lot of thinking about limitsand where they fit in the big picture of things. But I don’t lose sight of my goal.Įach year, I have parts of the calculus curriculum I rethink, or have insights on. Why? Because even though I could teach them that (and I have in the past), I would rather spend my time doing less work on moving through algebraic hoops, and more work on deep conceptual understanding.Įverything I do in my course aims for this. It is just that I pare things down so that they don’t have to find the derivatives of things like. Now don’t go thinking my kids come out of calculus not knowing how to do real calculus. My goal in this course is to get my kids to understand calculus with depth - that means my primary focus is on conceptual understanding, where facility with fancy-algebra things is secondary. Then the unique number, such obtained is called the left hand limit of f(x) at x = a.I’ve been meaning to write this post for a while. If values of the function at the points, very close to a on the left tends to a definite unique number as x tends to a. The limit value is having two types of values as Left-hand Limit and Right-Hand Limit. Also, we can see that a function value may or may not be the same as the limit and that either value may be undefined. The value of the function which is limited and can be different than the limit value of the function itself. Here, as x approaches 2, the limit of the function f(x) will be 5i.e. If these values tend to some definite unique number as x tends to a, then that obtained a unique number is called the limit of f(x) at x = a. If at a point x = a, f(x) takes indeterminate form, then we can consider the values of the function which is very near to a. 2 Solved Examples Limits Formula What is Limit?
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