2: Learning Limits
- Page ID
- 116543
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)The idea of a limit is central to all of calculus. We begin this chapter by examining why limits are so important. Then, we go on to describe how to find the limit of a function at a given point. Not all functions have limits at all points, and we discuss what this means and how we can tell if a function does or does not have a limit at a particular value. This chapter has been created in an informal, intuitive fashion, but this is not always enough if we need to prove a mathematical statement involving limits. The last section of this chapter presents the more precise definition of a limit and shows how to prove whether a function has a limit.
- 2.1: Tangent Lines and Velocity
- We begin our exploration of calculus by reconnecting with a topic from our early days in algebra - slope. The concept of slope is fundamentally important in calculus and this section, along with our old friend "slope," allows a gentle introduction to a monumentally important subject in mathematics and physics.
- 2.2: The Limit of a Function - A Numerical and Graphical Investigation
- This section explores the concept of the limit of a function through numerical and graphical approaches. It introduces the basic idea of limits, demonstrates how to estimate limits using tables of values, and examines behavior near a point using graphs. The section aims to build an intuitive understanding of limits as a foundational concept in Calculus, helping students visualize and analyze the approach of a function towards a particular value.
- 2.3: The Limit Laws - Limits at Finite Numbers
- This section introduces the Limit Laws for calculating limits at finite numbers. It covers fundamental rules, including the Sum, Difference, Product, Quotient, and Power Laws, which simplify finding limits of functions. The section emphasizes understanding and applying these laws systematically, providing examples and exercises to reinforce learning. These rules are essential for solving more complex limits and serve as a foundation for further study in Calculus.
- 2.4: The Limit Laws - Limits at Infinity
- This section discusses the limit laws for evaluating limits at infinity, focusing on the behavior of functions as they approach infinity or negative infinity. It covers rules for finding horizontal asymptotes and determining the end behavior of polynomial, rational, exponential, and logarithmic functions. Examples and exercises illustrate how to apply these laws in various contexts, helping to understand the asymptotic behavior of functions.
- 2.5: The Precise Definition of a Finite Limit at a Finite Number
- This section introduces the precise definition of a finite limit at a finite number using the epsilon-delta definition. It explains how to rigorously prove that a function approaches a particular limit as the input approaches a specific value. The section includes examples to demonstrate how to find appropriate delta values given an epsilon, helping to solidify the concept of limits in a formal mathematical context.
- 2.6: The Precise Definitions of Infinite Limits and Limits at Infinity
- This section provides the precise definitions of infinite limits and limits at infinity using formal epsilon-delta notation. It explains how to rigorously define what it means for a function to grow without bound or approach a value as the input tends to infinity. Examples illustrate the application of these definitions in different contexts, reinforcing the understanding of limits involving infinity.
- 2.7: Continuity
- This section introduces the concept of continuity in Calculus, explaining how a function is continuous at a point if the limit exists and equals the function's value at that point. It discusses the types of discontinuities (removable, jump, and infinite) and provides examples to illustrate these concepts. The section also covers the Intermediate Value Theorem, which relies on continuity to guarantee the existence of certain values within an interval.
- 2.8: Defining the Derivative
- This section defines the derivative using the limit process, focusing on the concept of the derivative as the slope of the tangent line or the instantaneous rate of change. It explains how to calculate the derivative through the limit of the difference quotient and provides practical examples of applying the derivative to functions. It also introduces notations and the foundational concept of differentiability.
- 2.9: The Derivative as a Function
- This section explains the derivative as a function, focusing on how the derivative varies across the domain of a function. It describes how the derivative itself can be viewed as a new function that gives the slope of the original function at any point. The section also covers the graphical interpretation, showing how the behavior of the derivative reflects changes in the original function's slope, such as increasing or decreasing trends.
Thumbnail: The function \(f(x)=1/(x−a)^n\) has infinite limits at \(a\). (CC BY; OpenStax)