1: Learning Limits — Read — Invisible Network

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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.

1.1: Numerical Investigation of Limits
This section discusses the concept of limits in calculus, emphasizing intuitive understanding before formal definitions. It covers estimating limits using tables, one-sided and two-sided limits, and the conditions for limits existing. Examples illustrate the behavior of functions as they approach specific points, including situations where limits do not exist. The importance of using radians for trigonometric functions and caution against over-reliance on technology are highlighted.
1.2: Graphical Investigation of Limits
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.
1.3: Limit Laws and the Squeeze Theorem
This section outlines methods for evaluating limits using Limit Laws and the Direct Substitution Property, covering basic rules and their application to polynomial and rational functions. It introduces the Squeeze Theorem for non-algebraic limits and emphasizes common student mistakes. Examples illustrate the evaluation of limits involving trigonometric functions, including foundational limits as \( \theta \to 0 \).
1.4: Algebraic Manipulations and the Limit Laws
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.
1.5: 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.
1.6: The Precise Definition of a Limit
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.
1.7: The Precise Definitions of Limits Involving 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.
1.8: 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.
1.9: 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.
1.10: 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.

This page titled 1: Learning Limits was last modified on Fri, 30 May 2025 22:40:17 GMT and is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Gilbert Strang & Edwin “Jed” Herman.

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