LC - Limits and Continuity Module Overview

Limits and Continuity Module Overview

Introduction

The notion of a limit arises naturally in our everyday existence. Colloquially, when we test or push a limit, the meaning is interpreted as determining just how far you can go before a consequence or limiting value occurs. This intuitive understanding of the limiting process is manifested in mathematics when we estimate limits from graphs and tables of data. A more formal approach to finding limits analytically uses algebraic techniques and properties of limits. One of the most important uses of limits in calculus is determining whether or not functions are continuous. Continuous functions are essential in modeling a large number of physical phenomena.

Essential Questions

What is a limit?
What role do limits play as a foundation for calculus and in practical application?
How can technology be used to find a limit with a table and a graph?
How is a limit found analytically?
When does a limit not exist?
How does the definition of continuity at a point relate to one-sided limits?
What is the definition of continuity of a function on an open or closed interval?
What does the Intermediate Value Theorem say about continuous functions and how does it relate to graphs of continuous functions?
What are infinite limits?
How are asymptotes reflected in graphical behavior and discontinuity?
How are asymptotic and unbounded behavior explained in terms of limits involving infinity?

Key Terms

The following key terms will help you understand the content in this module.

Limit of a function - A value L that function f(x) approaches increasingly closely as the independent variable, x, approaches a number a from either side of a but $LaTeX: x\ne a$ $x\ne a$ . This is denoted by $LaTeX: \lim_{x\to a }f\left(x\right) =L$ $\lim_{x\to a }f\left(x\right) =L$

One-sided limit - The limit where x is restricted to values less than a or greater than a. This is denoted by $LaTeX: \lim_ {x\rightarrow a^{-} }f\left(x\right) and \lim_ {x\rightarrow a^{+} }f\left(x\right)$ $\lim_ {x\rightarrow a^{-} }f\left(x\right) and \lim_ {x\rightarrow a^{+} }f\left(x\right)$

Continuous function - A function f(x) is continuous at a number a if f(a) is defined, $LaTeX: \lim_ {x\to a }f\left(x\right)$ $\lim_ {x\to a }f\left(x\right)$ exists, and $LaTeX: \lim_ {x\to a }f\left(x\right) =f(a)$ $\lim_ {x\to a }f\left(x\right) =f(a)$

Squeeze Theorem - If g(x) is squeezed between f(x) and h(x) near a, and if f and h have the same limit L at a, then g as the same limit L at a.

Intermediate Value Theorem - If f is a continuous function on the closed interval [a,b] and k is any. number between f(a) and f(b), then there exists a number c is (a, b) such that f(c) = k.

Discontinuous -If $LaTeX: \lim_{x\to a} f\left(x\right)\\$ $\lim_{x\to a} f\left(x\right)\\$ does not exist, or if it exists but does not equal f(a), then f is discontinuous at x=a.

Infinite discontinuity - A discontinuity at a specific value of c as it approached from one or both sides for which a function f approaches infiinty or negative infinity.

Removable discontinuity - A discontinuity at a specific value of c for a function f is removable if f can be made continuous by redefining f(c)

Jump discontinuity - If the left- and right-sided limits exist at x=a but disagree, then the graph of the function f(x) jumps at x=a.

Infinite limit - A limit where the value of a function f(x) increases or decreases without bound as x approaches a number a. If the values of f(x) increase without bound as x approaches a number a but $LaTeX: x\ne a$ $x\ne a$ , then $LaTeX: \lim_{x\to a} f\left(x\right)=\infty\\$ $\lim_{x\to a} f\left(x\right)=\infty\\$ . If the values of f(x) decreases without bound (large negative) as x approaches a number a, but $LaTeX: x\ne a$ $x\ne a$ , then $LaTeX: \lim_{x\to a}f\left(x\right)=-\infty\\$ $\lim_{x\to a}f\left(x\right)=-\infty\\$

Unbounded behavior - Having values of a function f(x) that increase or decrease without bound as x approaches a number a.

Oscillating behavior - Fluctuating between two fixed values of a function f(x) as x approaches a number a.

Asymptotic behavior - Becoming arbitrarily close to an asymptote as a function f(x) increases or decreases without bound.

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