Now we have a function that is continuous for all values of x. G(x) is continuous everywhere except at x=-1. The following function f_4(x) is continuous for all values of x: The figure below shows some examples, which are explained below: 3.1 The Square Function If the function is defined at a point, has no jumps at that point, and has a limit at that point, then it is continuous at that point. The concept of continuity is closely related to limits. Some examples follow: Examples of Continuity If all of the above hold true, then the function is continuous at the point a. The limit of f(x) is L as x approaches k, if for every □>0, there is a positive number □>0, such that: The mathematics community agrees to use □ for arbitrarily small positive numbers, which means we can make □ as small as we like and it can be as close to zero as we like, provided □>0 (so □ cannot be zero). To define a limit formally, we’ll use the the notion of the Greek letter □. In mathematics, we need to have an exact definition of everything. Therefore, this is another way of testing whether a function has a limit at a specific point, i.e, Formal Definition of a Limit We say that f(x) has a limit L as x approaches k, if both its left and right hand limits are equal. Both the left and right hand limits are written as follows: As we approach -1 from the right, the right hand limit of g(x) is 2. Similarly, the right hand limit is defined on an open interval to the right of -1 and does not include -1, e.g., (-1, 0.997). As we approach -1 from the left, g(x) gets closer to 2. The left hand limit is defined on an interval to the left of -1, which does not include -1, e.g., (-1.003, -1). This gives rise to the notion of one-sided limits. This is written as: Left and Right Hand Limitsįor the function g(x), it doesn’t matter whether we increase x to get closer to -1 (approach -1 from left) or decrease x to get closer to -1 (approach -1 from right), g(x) still gets closer and closer to 2. In general, for any function f(x), if f(x) gets closer and closer to a value L, as x gets closer and closer to k, we define the limit of f(x) as x approaches k, as L. If g(x) is defined in an open interval that does not include -1, and g(x) gets closer and closer to 2, as x approaches -1, we write this as: Despite the presence of this hole, g(x) gets closer and closer to 2 as x gets closer and closer -1, as shown in the figure: So it looks like there is a hole in the function at x=-1. However, at (x = -1), the denominator is zero and we cannot divide by zero. If the denominator is not zero then g(x) can be simplified as: We can simplify the expression for g(x) as: We say that f(x) has a limit equal to 0, when x approaches -1.Įxtending the problem. We can see that f(x) gets closer and closer to 0 as x gets closer and closer -1, from either side of x=-1. Let’s start by looking at a simple function f(x) given by: Photo by Mehreen Saeed, some rights reserved. In this post, you will discover how to evaluate the limit of a function, and how to determine if a function is continuous or not.Īfter reading this post, you will be able to:ĭetermine if a function f(x) has a limit as x approaches a certain valueĮvaluate the limit of a function f(x) as x approaches aĭetermine if a function is continuous at a point or in an intervalĭetermine if the limit of a function exists for a certain pointĬompute the limit of a function for a certain pointĭetermine if a function is continuous at a point or within an intervalĪ Gentle Introduction to Limits and Continuity The concept of limits and continuity serves as a foundation for all these topics. To understand machine learning algorithms, you need to understand concepts such as gradient of a function, Hessians of a matrix, and optimization, etc. However, if you learn the fundamentals, you will not only be able to grasp the more complex concepts but also find them fascinating. There is no denying that calculus is a difficult subject.
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