Problem 51 - Calculus Early Transcendentals: Pearson New International Edition

Question

Using the symbols \(M\) and \(\delta\), give precise definitions of each expression. (a) \(\lim _{x \rightarrow c^{+}} f(x)=-\infty\) (b) \(\lim _{x \rightarrow c^{-}} f(x)=\infty\)

Step-by-Step Solution

Answer

(a) For every \( M < 0 \), there exists \( \delta > 0 \) such that if \( 0 < x-c < \delta \), then \( f(x) < M \). (b) For every \( M > 0 \), there exists \( \delta > 0 \) such that if \( 0 < c-x < \delta \), then \( f(x) > M \).

Step 1: Understand the Given Notation

The notation \( \lim_{x \rightarrow c^+} f(x) = -\infty \) means that as \(x\) approaches \(c\) from the right, the function \(f(x)\) decreases without bound. Similarly, \( \lim_{x \rightarrow c^-} f(x) = \infty \) means that as \(x\) approaches \(c\) from the left, the function \(f(x)\) increases without bound.

Step 2: Define \\( \lim_{x \rightarrow c^+} f(x) = -\infty \\) Expression

The precise definition: For every real number \( M \), there exists a positive number \( \delta \) such that if \( 0 < x - c < \delta \), then \( f(x) < M \). This means that moving slightly right of \( c \), \( f(x) \) becomes less than any given negative number \( M \).

Step 3: Define \\( \lim_{x \rightarrow c^-} f(x) = \infty \\) Expression

The precise definition: For every real number \( M \), there exists a positive number \( \delta \) such that if \( 0 < c - x < \delta \), then \( f(x) > M \). This signifies that just to the left of \( c \), \( f(x) \) exceeds any given positive number \( M \).

Key Concepts

Unbounded Behavior

When we talk about unbounded behavior in calculus, we are describing functions that do not approach a finite number as the input value gets close to some point. Instead, the function's value either goes up to infinity or down to negative infinity. This is a key aspect of limits in calculus, especially when dealing with behavior near vertical asymptotes. - **Function going towards \(+\infty\)**: This means the outputs increase without limit. As input approaches a certain value, the functionality rises.- **Function going towards \(-\infty\)**: This indicates the outputs decrease without bound. As the input gets closer to a certain value, the functionality drops dramatically.When having unbounded behavior, it highlights how a function can become extremely large or small around certain points. Understanding this helps identify the "stretch" of a graph around those points, providing insight into its potential asymptotic nature.

Right-Hand Limit

A right-hand limit examines the behavior of a function as the input value approaches a specific point from the right. This type of limit is denoted by \( \lim_{x \rightarrow c^+} f(x) \). The `+` symbol indicates movement from values greater than \( c \). Here’s how this concept works in practice:- As \( x \) gets closer to \( c \) from the right, observe the values of \( f(x) \).- If \( f(x) \) decreases without bound (heads towards \(-\infty\)), how rapidly it decreases gives insight into the right-hand limit.A right-hand limit is particularly useful for functions with piecewise definitions, ensuring clarity on behavior when approached from the "right" side of a defined interval. The notation captures the direction of approach directly, and considering this can help identify divergence or convergence at unusual points within a function.

Left-Hand Limit

A left-hand limit explores the behavior of a function as the input value approaches from the left side of a specific point. In calculus, it is noted as \( \lim_{x \rightarrow c^-} f(x) \). The `-` symbol implies movement from values less than \( c \). Let's look at key components:- As \( x \) approaches \( c \) from the left, observe \( f(x) \).- If \( f(x) \) increases indefinitely (heads towards \(\infty\)), the rate of increase helps explain the left-hand limit.This notation is essential for proper understanding when discontinuities or drastic changes appear at specific values in a function. By focusing on such behavior, it equips learners to handle complex functions where the left-hand or right-hand values differ significantly.

Precise Definition of Limits

Precision in defining limits is crucial in calculus because it removes ambiguity. The use of specific values, like \(M\) and \(\delta\), provides a clear framework for understanding how closely a function approaches a boundary:- **Defining limits formally** involves \(\forall M, \exists \delta > 0\) for expressions like \(\lim_{x \rightarrow c^+} f(x) = -\infty\).- \(M\) represents any real number, and \(\delta\) signifies a positive distance from \(c\), ensuring \(f(x) \) is less than \(M\) when the domain condition is met.This precise language details how far values move across the graph and helps address infinite limits. By using this language, mathematicians and students communicate accurately about closeness and behavior near points of interest, reinforcing deeper understanding of calculus limits.