Post

Limits and Continuity

Created by @nathanedwards

at November 3rd 2023, 2:37:42 am.

AP_Calculus_AB-Overview

#calculus

#limits

#continuity

Limits and Continuity

Introduction

In calculus, the concept of limits and continuity plays a fundamental role in analyzing the behavior of functions. By understanding limits, we can determine how a function behaves as it approaches a certain point or as the input values approach infinity or negative infinity. Continuity, on the other hand, allows us to study the behavior of functions over a given interval. In this post, we will explore the concepts of limits and continuity, their properties, and the techniques used to evaluate them.

Limits

Definition

The limit of a function f(x) as x approaches a specific value c is denoted as:

\lim_{{x \to c}} f(x)

This limit represents the value that f(x) approaches as x gets arbitrarily close to c. It is important to distinguish between left-hand limits ( $x \to c^-$ ), right-hand limits ( $x \to c^+$ ), and two-sided limits.

Properties

Limits follow a set of properties that allow us to evaluate them algebraically. These properties include:

Sum/Difference Rule: The limit of the sum or difference of two functions is equal to the sum or difference of their limits. For example:

\lim_{{x \to c}} (f(x) \pm g(x)) = \lim_{{x \to c}} f(x) \pm \lim_{{x \to c}} g(x)

Product Rule: The limit of the product of two functions is equal to the product of their limits. Mathematically, we have:

\lim_{{x \to c}} (f(x) \cdot g(x)) = \lim_{{x \to c}} f(x) \cdot \lim_{{x \to c}} g(x)

Quotient Rule: The limit of the quotient of two functions is equal to the quotient of their limits, provided the denominator function is not zero. Symbolically, this can be expressed as:

\lim_{{x \to c}} \left(\frac{{f(x)}}{{g(x)}}\right) = \frac{{\lim_{{x \to c}} f(x)}}{{\lim_{{x \to c}} g(x)}}

Power Rule: The limit of a function raised to a constant power is equal to the limit of the function raised to that power. For instance:

\lim_{{x \to c}} (f(x))^n = (\lim_{{x \to c}} f(x))^n

Techniques for Evaluating Limits

The evaluation of limits often involves algebraic manipulation and application of several fundamental limit rules. Here are some commonly used techniques to evaluate limits:

Direct substitution: If substituting the value of c into the function does not result in an undefined expression (such as division by zero or square root of a negative number), then we can directly substitute and evaluate the limit.
Factoring and canceling: Sometimes, factoring the expressions involved or canceling common factors can help simplify the limit expression and make it easier to evaluate.
Rationalizing: Rationalization involves multiplying the numerator and denominator of a fraction by an appropriate conjugate (usually to eliminate square roots or radicals) to simplify the expression and evaluate the limit.
Squeeze theorem: The squeeze theorem is useful when we have a function trapped between two other functions with the same limit. In such cases, we can conclude that the function within the squeeze must also have that same limit.
L'Hôpital's Rule: L'Hôpital's Rule is applicable when we have an indeterminate form ( $\frac{0}{0}$ or $\frac{\infty}{\infty}$ ). It allows us to differentiate the numerator and denominator separately and then take the limit again.

Continuity

Definition

A function f(x) is said to be continuous at a point c if the following three conditions are met:

$f(c)$ is defined (the function is defined at c).
$\lim_{{x \to c}} f(x)$ exists (the limit of the function at c exists).
$\lim_{{x \to c}} f(x) = f(c)$ (the limit of the function at c is equal to the function value at c).

Types of Discontinuity

There are several types of discontinuity that can occur in a function. These include:

Removable Discontinuity: Also known as a "hole", this occurs when a function has a discontinuity at a point, but the function can be easily modified or filled in to make it continuous at that point.
Jump Discontinuity: A jump discontinuity happens when the left-hand limit and the right-hand limit of a function at a point exist, but they are not equal. This results in a "jump" in the function's values.
Infinite Discontinuity: An infinite discontinuity occurs when the limit of a function at a point approaches positive or negative infinity. This typically happens when the denominator of a fraction approaches zero or when a function has a vertical asymptote.

The Intermediate Value Theorem

The Intermediate Value Theorem states that if a function f(x) is continuous on a closed interval $[a, b]$ and $k$ is any value between $f(a)$ and $f(b)$ , then there exists at least one number $c$ in the interval $[a, b]$ such that $f(c) = k$ .

This theorem is useful for establishing the existence of roots or zeros for continuous functions. By evaluating the function at two endpoints of an interval and ensuring that they have different signs, we can conclude that the function has at least one root within that interval.

Conclusion

Limits and continuity are essential concepts in calculus that allow us to analyze and understand the behavior of functions. By evaluating limits, we can determine how a function approaches a certain value or behaves as the input values get infinitely large or small. Continuity, on the other hand, characterizes the smoothness and connectedness of a function. Understanding these concepts is crucial in many calculus applications, such as finding derivatives, solving equations, and studying the behavior of functions.