2: Learning Limits
- Page ID
- 116543
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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.
- 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
- A table of values or graph may be used to estimate a limit. If the limit of a function at a point does not exist, it is still possible that the limits from the left and right at that point may exist. If the limits of a function from the left and right exist and are equal, then the limit of the function is that common value. We may use limits to describe infinite behavior of a function at a point.
- 2.3: The Limit Laws - Limits at Finite Numbers
- In this section, we establish laws for calculating limits and learn how to apply these laws. We begin by restating two useful limit results from the previous section. These two results, together with the Limit Laws, serve as a foundation for calculating many limits.
- 2.4: The Limit Laws - Limits at Infinity
- In this section, we continue to investigate limits from an informal perspective. The limits focused on here are finite limits at infinity and infinite limits at infinity.
- 2.5: The Precise Definition of a Finite Limit at a Finite Number
- In this section, we convert this intuitive idea of a limit into a formal definition using precise mathematical language. The formal definition of a limit is quite possibly one of the most challenging definitions you will encounter early in your study of calculus; however, it is well worth any effort you make to reconcile it with your intuitive notion of a limit. Understanding this definition is the key that opens the door to a better understanding of calculus.
- 2.6: The Precise Definitions of Infinite Limits and Limits at Infinity
- In this section, we enter the second half of our discussion of the precise definitions of limits. We focus on infinite limits and limits at infinity.
- 2.7: Continuity
- For a function to be continuous at a point, it must be defined at that point, its limit must exist at the point, and the value of the function at that point must equal the value of the limit at that point. Discontinuities may be classified as removable, jump, or infinite. A function is continuous over an open interval if it is continuous at every point in the interval. It is continuous over a closed interval if it is continuous at every point in its interior and is continuous at its endpoints.
- 2.8: Defining the Derivative
- The slope of the tangent line to a curve measures the instantaneous rate of change of a curve. We can calculate it by finding the limit of the difference quotient or the difference quotient with increment h . The derivative of a function f(x) at a value a is found using either of the definitions for the slope of the tangent line. Velocity is the rate of change of position. As such, the velocity v(t) at time t is the derivative of the position s(t) at time t .
- 2.9: The Derivative as a Function
- The derivative of a function f(x) is the function whose value at x is f′(x). The graph of a derivative of a function f(x) is related to the graph of f(x). Where (f(x) has a tangent line with positive slope, f′(x)>0. Where (x) has a tangent line with negative slope, f′(x)<0. Where f(x) has a horizontal tangent line, f′(x)=0. If a function is differentiable at a point, then it is continuous at that point.
Thumbnail: The function \(f(x)=1/(x−a)^n\) has infinite limits at \(a\). (CC BY; OpenStax)