If the function is not continuous, even if it is defined, at a particular point, then the limit will not necessarily be the same value as the actual function. In fact, that is one way of defining a continuous function:į(c) = lim x→c⁻ f(x) = lim x→c⁺ f(x) for all values of c within the domain.īut, suppose that there is something unusual that happens with the function at a particular point. If you have a continuous function, then this limit will be the same thing as the actual value of the function at that point. Many aspects of calculus also have geometric interpretations in terms of areas, slopes, tangent lines, etc.Ī limit is a method of determining what it looks like the function "ought to be" at a particular point based on what the function is doing as you get close to that point. The reason you see a lot of, say, algebra in calculus, is because many of the definitions in the subject are based on the algebraic structure of the real line. Some calculus courses focus most on the computational aspects, some more on the theoretical aspects, and others tend to focus on both. Elementary calculus is also largely concerned with such questions as how does one compute the derivative of a differentiable function? How does one compute the integral of an integrable function? Here there are many techniques to be mastered, e.g., the product rule, the chain rule, integration by parts, change of variable in an integral. One should regard these theorems as descriptions of the various classes.Īnd then there is, of course, the computational aspect. If one knows that a function ƒ is continuous, what else can you say about ƒ? The intermediate value theorem, the extreme value theorem, and so on, are examples of theorems describing further properties enjoyed by continuous functions. For instance, an integrable function may be less smooth (in some appropriate sense) than a continuous function, which may be less smooth than a differentiable function, which may be less smooth than a twice differentiable function, and so on. Some insight will reveal that this process of grouping functions into classes is an attempt to categorize functions with respect to how "smooth" or "well-behaved" they are. Examples of such classes are the continuous functions, the differentiable functions, the integrable functions, etc. One divides these functions into different classes depending on their properties. Want to work at KA? We hire interns too!įor learning languages, we love Duolingo and Memrise.Elementary calculus may be described as a study of real-valued functions on the real line.More blogs: Life at KA, School implementations, Schools blog, KA International. Official KA blog, Twitter account and Facebook page.If you have a question about a specific video, you can ask below the video itself. To report a technical problem or learn more about Khan Academy, you can visit the Help Center. To get in contact with us, see the Help Center or our Press page. This subreddit is run by Khan Academy employees, and we try to answer questions when we can, but keep in mind that it shouldn't be seen as an official way of contacting us. Khan Academy is a not-for-profit with the goal of providing a free world-class education for anyone anywhere. This is a forum for discussion about Khan Academy, and learning in general.
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