![]() $$f(x)$$ is continuous on the closed interval $$$$ if it is continuous on $$(a,b)$$, and one-sided continuous at each of the endpoints. With one-sided continuity defined, we can now talk about continuity on a closed interval. One-sided continuity is important when we want to discuss continuity on a closed interval. Definition: $$\displaystyle\lim\limits_ f(x) = f(a)$$.Note that \(y\) is undefined at \(x=1\), but the limit at \(x=1\) is defined (it is 3, what would have been the \(y\)-value without the hole). Here’s a graphical example of a removable discontinuity, or hole, that represents a limit at the \(x\)-value where the discontinuity exists. As an example, when you first learn how to handle limits, it might be the case that the \(x\)-value is getting closer and closer to a number that makes the denominator of the \(y\)-value 0 this would typically be undefined. Sometimes, the \(x\)-value does get there (like when we’re taking the slope of a straight line), but sometimes it doesn’t (like when we’re taking the slope of a curved function). ![]() The reason we have limits in Differential Calculus is because sometimes we need to know what happens to a function when \(x\) gets closer and closer to a number (but doesn’t actually get there) we will use this concept in getting the approximation of a slope (“rate”) of a curve at that point. Even if a function is “normal”, like a linear function, we still consider the \(y\)-value a limit where it touches the \(x\)-value, as shown below. Now the beauty of limits is that \(x\) can get closer and closer to a number, but not actually ever get there (think asymptote from the Rational Functions section). ![]() We can write a limit where \(x\) gets closer and closer to 0 as \(\underset\)”. Again, remember that limits are always the \(y\)-value (dependent variable), not the \(x\)-value (independent variable). (Differential calculus has to do with rates at which quantities change.) I like to think of a limit as what the \(y\)-part of a graph or function approaches as \(x\) gets closer and closer to a number, either from the left-hand side (which means that \(x\)-part is increasing), or from the right-hand side (which means the \(x\)-part is decreasing). We need to understand how limits work, since the first part of Differential Calculus uses them extensively. Also note that there’s a very good limit calculator here on the this online calculator site. Note that we discuss finding limits using L’Hopital’s Rule here. Introduction to Limits Intermediate Value Theorem (IVT) Finding Limits Algebraically Infinite Limits Continuity and One Side Limits Limits at Infinity Continuity of Functions Limits of Sequences Properties of Limits More Practice Limits with Sine and Cosine
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