If we make the denominator smaller, the value of the fraction gets larger: = 500, = 500000. divide 5 by 0.1, we get that it equals 50. For example, if we look at the fraction, i.e. We look for asymptotes at points where the denominator is zero because: when the denominator gets close to zero and becomes very small, it makes the value of the function very large. If a function has values on both sides of an asymptote, then it cannot be connected, so it must have a discontinuity at the asymptote. There are three types of discontinuities: Discontinuity 1: Asymptotic DiscontinuitiesĪsymptotes occur when a function approaches infinity at a specific value of x or y. Now, what would a discontinuous function look like? A function essentially is discontinuous when it has any “gap”. The most important is to recognize a continuous function when you see it. If you are confused by that, ignore it! You don’t need to learn all at once. There is a precise mathematical definition of continuity that uses limits. Intuitively, this definition says that small changes in the input of the function result in small changes in the output. Here’s an example of what a continuous function looks like: The basic idea of continuity is very simple, and the “formal” definition uses limits.īasically, we say a function is continuous when you can graph it without lifting your pencil from the paper. Limits and continuity are often covered in the same chapter of textbooks. The value of the function at the specific point we care about is not defined, like 0/0 (which is complete junk), or useless, like zero or infinite. In calculus, the most useful limits are like this one. What does the function approach when x approaches 1? It also approaches 1, right? It doesn’t matter that the function is other than 1 at that point! So, This function is the same as the one we saw before, but in this case it has a “hole” at x=1. Well, the point is that sometimes we don’t care what the function is at x=1.Īs an example of this, let’s consider the following function:ĭon’t let this notation intimidate you! This only means that this function equals x 2 when x is anything other than 1, and equals 0 when x equals 1. Why would you need to know what the function is approaching? You already know the function equals 1 when x equals 1, right? This is read “the limit as x approaches 1 of x squared equals 1”. We can see from the graph that when x approaches 1, the function f(x) approaches 1. Let’s turn to the graph of a function whose expression we know: I do this because we don’t necessarily know the value of function f at x=a. In this graph I drawed a big pink hole at the point (a,L). When x approaches the value “a” in the x axis, the function f(x) approaches “L” in the y axis. To start getting used to this idea, let’s turn to this graph: The idea behind limits is to analyze what the function is “approaching” when x “approaches” a specific value. For example, the function y = x 2 + 2 assigns the value y = 3 to x = 1, y = 6to x = 2, and y = 11 to x = 3. Using this function, we can generate a set of ordered pairs of ( x, y) including (1, 3),(2, 6), and (3, 11). We all know about functions, A function is a rule that assigns to each element xfrom a set known as the “ domain” a single element yfrom a set known as the “ range“. With an understanding of the concepts of limits and continuity, you are ready for calculus. Using limits, we’ll learn a better and far more precise way of defining continuity as well. For the math that we are doing in precalculus and calculus, a conceptual definition of continuity like this one is probably sufficient, but for higher math, a more technical definition is needed. One easy way to test for the continuity of a function is to see whether the graph of a function can be traced with a pen without lifting the pen from the paper. A function can either be continuous or discontinuous. ContinuityĬontinuity is another far-reaching concept in calculus. ” Symbolically, this is written f ( x) = 6. For example, given the function f ( x) = 3 x, you could say, “The limit of f ( x) as xapproaches 2 is 6. LimitĪ limit is a number that a function approaches as the independent variable of the function approaches a given value. We still use the Leibniz notation of dy/dx for most purposes. These two gentlemen are the founding fathers of Calculus and they did most of their work in 1600s. Gottfried Leibnitz is a famous German philosopher and mathematician and he was a contemporary of Isaac Newton. The concept of the Limits and Continuity is one of the most crucial things to understand in order to prepare for calculus.
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