Problem 59 - Calculus Early Transcendentals: Pearson New International Edition

Question

Since calculus software packages find \(\lim _{x \rightarrow a} f(x)\) by sampling a few values of \(f(x)\) for \(x\) near \(a\), they can be fooled. Find a function \(f\) for which \(\lim _{x \rightarrow 0} f(x)\) fails to exist but for which your software gives a value for the limit.

Step-by-Step Solution

Answer

Use the function \(f(x) = \sin\left(\frac{1}{x}\right)\); the limit does not exist, but software might return a value.

Step 1: Problem Understanding

We need to find a function for which the limit as \(x\) approaches \(0\) does not exist, but a calculus software might still return a limit value. This usually happens with piecewise or discontinuous functions.

Step 2: Select a Function

Consider the piecewise function \(f(x) = \begin{cases} \sin\left(\frac{1}{x}\right) & \text{if } x eq 0, \ 0 & \text{if } x = 0. \end{cases}\). This function is notorious for oscillating infinitely as \(x\) approaches \(0\).

Step 3: Analyze Limit Behavior

Analyze \( \lim_{x \to 0} \sin\left(\frac{1}{x}\right) \). As \(x\) approaches \(0\), the frequency of \(\sin\left(\frac{1}{x}\right)\) increases, causing the values to oscillate between \(-1\) and \(1\) infinitely. Hence, the limit does not exist because the values do not approach a single number.

Step 4: Understand Software Approximation

Calculus software often samples a few points around \(x = 0\). If those sampled points coincide in a particular sector of \(\sin\left(\frac{1}{x}\right)\), the software might mistakenly infer a limit value, such as zero or any other value.

Key Concepts

Discontinuous Functions

A discontinuous function is one whose output does not smoothly follow its input, meaning there are jumps or breaks in the graph. This often occurs at specific points where the function is not defined in a traditional sense. At these points of discontinuity, the behavior of the function becomes unpredictable or chaotic.

In the world of calculus, discontinuous functions can pose challenges when calculating limits. For example, consider a function that oscillates or behaves erratically as it nears a certain value. This might affect the limit, making it fail to exist at that point.

In our exercise, the chosen function involved trigonometric oscillation and specifically exhibited discontinuity at a critical point that challenged limit calculation.

Piecewise Functions

A piecewise function is composed of multiple sub-functions, each with its own specific formula and defined for a particular interval in the domain. These sub-functions can be continuous or discontinuous, depending on how they are defined at the boundaries between intervals.

Take the example from the exercise, where the function is defined as follows:

• For nonzero values of \(x\), the function is \(f(x) = \sin\left(\frac{1}{x}\right)\).
• For \(x = 0\), the function assigns a value directly, making it \(f(x) = 0\).

This discontinuity creates an intriguing situation where the function swings between different values for every tiny change in \(x\). Thus, evaluating limits can become complex, as seen when the software attempts calculations and samples only a finite set of points.

Limit Behavior Analysis

Limit behavior analysis involves examining how a function behaves as it approaches a certain point. It is critical to understanding concepts in calculus like convergence or divergence.

Analyzing the limit of the function \(f(x) = \sin\left(\frac{1}{x}\right)\) presents a unique challenge. As \(x\) approaches zero, the frequency of oscillations increases dramatically, causing the function values to vary rapidly between \(-1\) and \(1\). This indicates the limit does not exist at \(x = 0\) because there's no single value that the function settles into.

However, software might return a limit due to the way it samples points. If the sampled points happen to average around a particular value, like zero, the software may incorrectly report this as the limit, obscuring the true behavior of the function.