Continuity of a function at a point and on an interval will be defined using limits. If r and s are integers, s 0, then lim xc f x r s lr s provided that lr s is a real number. Functions, limits, continuity this module includes chapter p and 1 from calculus by adams and essex and is taught in three lectures, two tutorials and one seminar. Using this definition, it is possible to find the value of the limits given a graph. Let be a function defined on an open interval containing c except possibly at c and let l be a real number. In this chapter, we want to look at functions on metric. Lecture note 2 norwegian university of science and technology professor jon kleppe department of petroleum engineering and applied geophysics 11. Overview this lecture introduces much of the terminology we will use in the course, and we will describe it in more detail later.
Benginning calculus lecture notes 2 limits and continuity 1. Limits and continuity i lecture notes free download as pdf file. A function f is continuous at x 0 if lim x x 0 fx fx 0. C is a rule that assigns unique complex number, denoted by fz to every number z2s. The following is an immediate consequence of the a. To prove a limit doesnt exist, find two paths to a,b that give different limit values.
The work of hilbert and his pupils in gottingen on integral equations, spectral theory, and. Calculus for the life sciences i lecture notes limits, continuity. Analysis based on continuity 1 this lecture presents several techniques of qualitative systems analysis based on what is frequently called topological arguments, i. Limits, continuity, and differentiability student sessionpresenter notes this session includes a reference sheet at the back of the packet since for most students it has been some time since they have studied limits. From there, i earned my undergraduate degree from plymouth state college and my masters degree from new england college. Limits and continuity in the last section, we saw that as the interval over which we calculated got smaller, the secant slopes approached the tangent slope. Chapter 5 functions on metric spaces and continuity when we studied realvalued functions of a real variable in calculus, the techniques and theory built on properties of continuity, differentiability, and integrability. Limits and continuity these revision exercises will help you practise the procedures involved in finding limits and examining the continuity of functions.
Then 3x4 is very large, and also positive because it is the product of ve positive. Lecture 3 limits and derivatives university of toronto. The latex and python les which were used to produce these notes are available at the following web site. We say the limit of fx as x approaches a is l, denoted lim x. The limit gives us better language with which to discuss the idea of approaches. Current density and the continuity equation current is motion of charges. Continuity the conventional approach to calculus is founded on limits. These are some notes on introductory real analysis. Categories and functors lecture notes for midlands graduate. The limit of a function describes the behavior of the function when the variable is. To develop a useful theory, we must instead restrict the class of functions we consider. Both of these examples involve the concept of limits, which we will investigate in this module. The limit of a rational power of a function is that power of the limit of the function, provided the latter is a real number.
We say that the limit of fx as x tends to c is l and write lim xc. Rhas a discontinuous graph as shown in the following. Limits and continuity algebra reveals much about many functions. We do not mean to indicate that we are actually dividing by zero. Continuity in this section we will introduce the concept of continuity and how it relates to limits. In this chapter, we will develop the concept of a limit by example. The notes were written by sigurd angenent, starting from an extensive collection of notes and problems compiled by joel robbin. Categories and functors lecture notes for midlands graduate school, 2012 uday s. In em, we are often interested in events at a point. Limits and continuity i lecture notes limit mathematics function. Let f be a function of two variables whose domain d contains points arbitrarily close to the point a,b. We have sometimes stated that there is division by zero.
The cost at the gas station depends on the number of gallons of gas you pumped into your car. Lecture 3 notes these notes correspond to section 11. Lecture 1 for the ap calculus summer assignment at poteet high school. However, the continuity equation for gas has to be modified to include solution gas as well as free gas, so that the. We will use limits to analyze asymptotic behaviors of functions and their graphs. However, there are places where the algebra breaks down thanks to division by zero. Limits of functions and continuity kosuke imai department of politics, princeton university october 18, 2005 in this chapter, we study limits of functions and the concept of continuity. Persistence of solutions we discuss the persistence of solutions to nonlinear equations. All these topics are taught in math108, but are also needed for math109. Benginning calculus lecture notes 2 limits and continuity.
Trench, introduction to real analysis free online at. Now that we have a good understanding of limits of sequences, it should not be too di. I am a proud graduate of merrimack valley high school class of 88. Limit and continuity of functions ra kul alam department of mathematics iit guwahati ra kul alam ma102 20. This section contains lecture video excerpts, lecture notes, a worked example, a problem solving video, and an interactive mathlet. Limits and continuity of various types of functions. Here is a set of assignement problems for use by instructors to accompany the continuity section of the limits chapter of the notes for paul dawkins calculus i course at lamar university. Limits and continuity in this section, we will learn about.
A function f is continuous at a if to verify continuity, we need to check three things. Lecture notes limits at innity part 1 page 4 sample problems solutions 1. Ap calculus limits, continuity, and differentiability. Here are my online notes for my calculus i course that i teach here at lamar university. Nov 03, 2014 benginning calculus lecture notes 2 limits and continuity 1. Beginning calculus limits and continuity shahrizal shamsuddin norashiqin mohd idrus department of mathematics, fsmt upsi lecture slides series villarino domath, fsmtupsi d1 limits and continuity 1 54. In general, you can see that these limits are equal to the value of the function. The amount of your paycheck if you are paid hourly depends on the number of hours you worked. Intuitively, a function is continuous if you can draw its graph without picking up your pencil. For more information and feedback, visit out website. This session discusses limits and introduces the related concept of continuity. In addition, for differentiable functions well explore a variety of results growing.
Lecture notes on numerical analysis of of nonlinear equations eusebius doedel 1. Continuity and limits intuitively, we think of a function f. Landsman kortewegde vries institute for mathematics, university of amsterdam. We say the limit of fx,y as x,y approaches a,b within the domain d is the number l and we. This section considers some examples of phenomena where limits arise in a natural way. The limits are defined as the value that the function approaches as it goes to an x value. May 27, 2017 lecture 1 for the ap calculus summer assignment at poteet high school. We will also see the mean value theorem in this section. A function thats continuous at x 0 has the following properties. Beginning calculus limits and continuity shahrizal shamsuddin norashiqin mohd idrus department of mathematics, fsmt upsi lecture slides series villarino domath, fsmtupsi. Theorem 2 polynomial and rational functions nn a a.
Common types of behavior associated with nonexistence of a limit. This section contains lecture video excerpts, lecture notes, a worked example, a problem solving video, and an interactive mathlet with supporting documents. In mathematics we often describe one quantity in terms of another. N02 is the normal distribution with mean 0 and variance. Chapter 5 functions on metric spaces and continuity. Categories and functors lecture notes for midlands. The formal definition of a limit is generally not covered in secondary.
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