We have also included a limits calculator at the end of this lesson. Limits of polynomials and rational functions if f is a polynomial function, then lim x a f x exists and is given by lim x a f x f a an important limit an important limit which is very useful and used in the sequel is given below. This has the same definition as the limit except it requires xa pdf these are some lecture notes for the calculus i course. Limits, derivatives, applications of derivatives, basic integration revised in fall, 2018.
I may keep working on this document as the course goes on, so these notes will not be completely. The divisions into chapters in these notes, the order of the chapters, and the order of items within a chapter is in no way intended to re ect opinions i have about the way in which or even if calculus should be taught. In general, a fractional function will have an infinite limit if the limit of the denominator is zero and the limit of the numerator is not zero. For each of the given points determine the value of f. Provided by the academic center for excellence 4 calculus limits. Many expressions in calculus are simpler in base e than in other bases like base 2 or base 10 i e 2.
This section contains lecture video excerpts, lecture notes, a worked example, a problem solving video, and an interactive mathlet with supporting documents. These techniques include factoring, multiplying by the conjugate. Understanding basic calculus graduate school of mathematics. The following table gives the existence of limit theorem and the definition of continuity. We will also give a brief introduction to a precise definition of the limit and how to use it to. Continuity and discontinuity 3 we say a function is continuous if its domain is an interval, and it is continuous at every point of that interval. There are many more kinds of indeterminate forms and we will be discussing indeterminate forms at length in the next chapter. Lecture notes single variable calculus mathematics mit. Math 221 1st semester calculus lecture notes version 2. Pdf these are some lecture notes for the calculus i course.
To find the limit as approaches 1 from the left side, the first equation must be used because it defines the function at values less than and equal to one. Provided by the academic center for excellence 1 calculus limits november 20 calculus limits images in this handout were obtained from the my math lab briggs online ebook. Ive tried to make these notes as self contained as possible and so all the information needed to read through them is either from an algebra or trig class or contained in other sections of the notes. The notes were written by sigurd angenent, starting from an extensive collection of notes and problems compiled by joel robbin. In this chapter, we will develop the concept of a limit by example.
Limits are used to define continuity, derivatives, and integral s. This session discusses limits and introduces the related concept of continuity. Basic limit theorem for rational functions if f is a rational function, and a domf, then lim x a. Draft calculus notes 11172011 9 preface these notes are being written for an introductory honors calculus class, math 1551, at lsu in the fall of 2011. To evaluate the limits of trigonometric functions, we shall make use of the following limits which are.
For applications will take problems from various texts such as. Let f be a function defined at each point of some open interval containing a, except possibly a itself. Then a number l is the limit of f x as x approaches a or is the limit of. In other words, limits in which the variable gets very large in either the positive or negative sense. Calculus i or needing a refresher in some of the early topics in calculus. It was developed in the 17th century to study four major classes of scienti. Erdman portland state university version august 1, 20. Infinite limits here we will take a look at limits that have a value of infinity or negative infinity. Be sure to get the pdf files if you want to print them. Accompanying the pdf file of this book is a set of mathematica. However, there are also many limits for which this wont work easily.
Benginning calculus lecture notes 2 limits and continuity. A point of discontinuity is always understood to be isolated, i. Notes on calculus ii integral calculus nu math sites. Make sure your calculator is set to radians for the computations. There are videos pencasts for some of the sections. Special limits e the natural base i the number e is the natural base in calculus. Calculus 1 class notes, thomas calculus, early transcendentals, 12th edition copies of the classnotes are on the internet in pdf format as given below. Because i want these notes to provide some more examples for you to read. Examples are methods such as newtons method, fixed point iteration, and linear approximation. Historically, two problems are used to introduce the basic tenets of calculus. This has the same definition as the limit except it requires xa limit at infinity. I e is easy to remember to 9 decimal places because 1828 repeats twice.
The concept of a limit of a function is essential to the study of calculus. This book is a revised and expanded version of the lecture notes for basic calculus and. But you can say that as you approach 1, the limit is 2. This book is a revised and expanded version of the lecture notes for basic calculus and other similar courses o ered by the department of mathematics, university of hong kong, from the. Use the information from a to estimate the value of lim. 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 2. So, in truth, you cannot say what the value at x1 is. Calculus cheat sheet limits pauls online math notes. In this section our approach to this important concept will be intuitive, concentrating on understanding what a limit is using numerical and. Assume dont know f x0 but know fx for any x close to x0. Lecture notes on the lambda calculus pdf 106p this notes contains the details about the untyped lambda calculus, the churchrosser theorem, combinatory algebras, the curryhoward isomorphism, polymorphism, weak and strong normalization, denotational semantics of pcf. 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. It deals with fundamental limits first and the rules of differentiation for all the.
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. In these notes we would like to take an approach that goes to the matter of things most of the time. Properties of limits will be established along the way. It is used in defining some of the more important concepts in calculus. Nov 03, 2014 benginning calculus lecture notes 2 limits and continuity 1. There is a similar definition for lim x fxl except we requirxe large and negative. Note also that the function has a vertical asymptote at x c if either of the above limits hold true. This property is crucial for calculus, but arguments using it are too di cult for an introductory course on the subject. In general, any limit of the six types 3 is called an in. I havent written up notes on all the topics in my calculus courses, and some of these notes are incomplete they may contain just a few examples, with little exposition and few proofs. However, in take the limit, if we get 00 we can get a variety of answers and the only way to know which on is correct is to actually compute the limit. A limit is the value a function approaches as the input value gets closer to a specified quantity. You may feel embarrassed to nd out that you have already forgotten a number of things that you learned di erential calculus.
Calculus limits of functions solutions, examples, videos. Theorem 2 polynomial and rational functions nn a a. In the previous section we saw that there is a large class of functions that allows us to use. Notes on first semester calculus singlevariable calculus. Multivariable calculus before we tackle the very large subject of calculus of functions of several variables, you should know the applications that motivate this topic. This math tool will show you the steps to find the limits of a given function. These notes are intended to be a summary of the main ideas in course math 2142. Our study of calculus begins with an understanding of the expression lim x a fx. Calculus i computing limits pauls online math notes.
Remark the above expression remains valid for any rational number provided a is. We look at a few examples to refresh the readers memory of some standard techniques. In fact if i had to choose a subtitle for these notes, it would be an anticalculustext book. Functions and their graphs, trigonometric functions, exponential functions, limits and continuity, differentiation, differentiation rules, implicit differentiation, inverse trigonometric functions, derivatives of inverse functions and logarithms, applications of derivatives, extreme values of functions, the mean value theorem.
In this chapter we introduce the concept of limits. When this occurs, the function is said to have an infinite limit. The approach is quite di erent from that of standard calculus texts. Well also take a brief look at vertical asymptotes. We say lim x fxl if we can make fx as close to l as we want by taking x large enough and positive. These notes are written for a onesemester calculus course which meets three times a week and is, preferably, supported by a computer lab. Cisnero, ap calculus bc chapter 1 notes as a graph it looks like this. The course is designed for life science majors who have a precalculus back ground, and whose primary interest lies in the applications of calculus. Calculus is also used to find approximate solutions to equations. Pdf produced by some word processors for output purposes only. Lecture notes on integral calculus ubc math 103 lecture notes by yuexian li spring, 2004 1 introduction and highlights di erential calculus you learned in the past term was about di erentiation.
Lecture notes single variable calculus mathematics. We will discuss the interpretationmeaning of a limit, how to evaluate limits, the definition and evaluation of onesided limits, evaluation of infinite limits, evaluation of limits at infinity, continuity and the intermediate value theorem. The idea of limit can be intuitively given by some important. Again, in each case of 3 we are simply describing in a symbolic manner the behavior of a function fnear the number none of the limits ina. Limits at infinity, part i in this section well look at limits at infinity. The conventional approach to calculus is founded on limits. They dont cover all the material in the printed notes the web pages and pdf files, but i try to hit the important points and give enough examples to get you started. The limit of a function fx describes the behavior of the function close to a particular x value.
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