Problem 58 - Essential Calculus Early Transcendentals

Question

$$f(x)=\left\\{\begin{array}{ll}{|x|^{x}} & {\text { if } x \neq 0} \\ {1} & {\text { if } x=0}\end{array}\right.$$ (a) Show that \(f\) is continuous at 0 . (b) Investigate graphically whether \(f\) is differentiable at 0 by zooming in several times toward the point \((0,1)\) on the graph of \(f .\) (c) Show that \(f\) is not differentiable at \(0 .\) How can you reconcile this fact with the appearance of the graphs in part (b)?

Step-by-Step Solution

Answer

(a) \(f\) is continuous at 0. (b) Graphically, it may appear differentiable. (c) Analytically, \(f\) is not differentiable at 0 due to oscillations.

Step 1: Define Continuity at a Point

To show that a function \(f(x)\) is continuous at a point \(x = c\), we need to show that \(\lim_{x \to c} f(x) = f(c)\). For the given function, we need to check whether \(\lim_{x \to 0} f(x) = 1\).

Step 2: Explore Limiting Behavior of \(f(x)\) as \(x\) Approaches 0

When \(x eq 0\), the function \(f(x) = |x|^x = e^{x\ln|x|}\). To find the limit, evaluate \(\lim_{x \to 0} e^{x\ln|x|}\). This requires investigating \(\lim_{x \to 0} x\ln|x|\).

Step 3: Calculate \(\lim_{x \to 0} x\ln|x|\) Using L'Hôpital's Rule

First, check the limit of \( x \ln|x| \) as \(x\) approaches 0. Rewrite this as \(\frac{\ln|x|}{1/x}\). Applying L'Hôpital's Rule due to the form \(\frac{-\infty}{\infty}\), compute derivatives of numerator and denominator to get \(\frac{1/|x|}{-1/x^2} = -x\) which tends to 0 as \(x\) approaches 0.

Step 4: Conclude Limit of \(e^{x\ln|x|}\)

Since \(\lim_{x \to 0} x\ln|x| = 0\), therefore \(\lim_{x \to 0} e^{x\ln|x|} = e^0 = 1\). Thus, \(\lim_{x \to 0} f(x) = 1\). Hence, \(f\) is continuous at \(x = 0\) since the limit equals \(f(0)\).

Step 5: Graphical Investigation for Differentiability

To investigate differentiability graphically, zoom in on the graph of \(f(x)\) around the point (0,1). Observe if the graph becomes a straight line as you zoom in closer, which suggests a derivative exists.

Step 6: Analyze the Appearance of the Graph

Upon zooming in repeatedly at the point \((0, 1)\), the graph of \(f(x)\) may appear to become more linear, but this visual inspection isn't conclusive proof of differentiability.

Step 7: Show Non-Differentiability Analytically

To show \(f\) is not differentiable at \(x = 0\), we calculate the derivative using the limit definition: \(f'(0) = \lim_{h \to 0} \frac{f(h) - f(0)}{h} = \lim_{h \to 0} \frac{|h|^h - 1}{h}\). Evaluate \(\lim_{h \to 0} \frac{e^{h \ln |h|} - 1}{h}\). Since \(h\ln|h|\) tends to 0 but not differentiably (oscillations and undefined nature of logarithm near 0), this limit does not exist.

Step 8: Reconcile Graphical Appearance with Analytical Results

The oscillating nature of \(x\ln|x|\) near 0 suggests non-uniform behavior that geometric zooming might not capture. The smooth appearance in a graph due to limited resolution fails to reveal these oscillations, which leads to non-differentiability.

Key Concepts

Continuity at a Point

Understanding continuity at a point is essential for analyzing how functions behave as they approach specific values. When we say a function is continuous at a point, it means there are no abrupt jumps or breaks around that point. Mathematically, for a function \( f(x) \) to be continuous at a point \( x = c \), the following must hold true:

• The function \( f(c) \) must be defined.
• The limit \( \lim_{x \to c} f(x) \) must exist.
• The limit \( \lim_{x \to c} f(x) \) must equal the function's value \( f(c) \).

In the case of the function \( f(x) = |x|^x \) where \( x eq 0 \) and \( f(0) = 1 \), demonstrating continuity at \( x = 0 \) involves showing that \( \lim_{x \to 0} f(x) = 1 \). By analyzing the limiting behavior of \( e^{x \ln |x|} \), and confirming it converges to 1, it affirms that the function is indeed continuous at \( x = 0 \). This simple act ensures a smooth transition in function behavior across the point.

Graphical Investigation

Graphical investigation helps visually ascertain the behavior of functions near particular points, providing intuition about continuity and differentiability. In this context, investigating the point \((0,1)\) for the function \( f(x) = |x|^x \) involves zooming in repeatedly on the graph.
This gradual zooming allows us to observe whether the graph appears to straighten, indicating potential differentiability. As we hone in closer to \((0,1)\), the graph may seem linear due to the decreasing effect of higher-order terms that obscure the local behavior initially.
However, graphical methods are limited since they rely on visual resolution and perception. They offer initial insights, encouraging a closer analytical look for definitive conclusions.

L'Hôpital's Rule

L'Hôpital's Rule provides a powerful technique for evaluating limits of indeterminate forms like \( \frac{0}{0} \) or \( \frac{-\infty}{\infty} \). This rule states that for functions \( f(x) \) and \( g(x) \), if \( \lim_{x \to c} \frac{f(x)}{g(x)} \) takes an indeterminate form, the limit can be evaluated as:

• If \( \lim_{x \to c} f'(x) \) and \( \lim_{x \to c} g'(x) \) exist or approach infinity.
• \( \lim_{x \to c} \frac{f(x)}{g(x)} = \lim_{x \to c} \frac{f'(x)}{g'(x)} \).

In examining \( \lim_{x \to 0} x \ln |x| \), we rewrite it as \( \frac{\ln |x|}{1/x} \) and identify the forms necessitating L'Hôpital's Rule. Derivatives are then taken to simplify the evaluation, resolving the limit to 0. This facilitates understanding the continuous behavior of \( f(x) = |x|^x \) near zero.

Limit Definition of Derivative

The limit definition of a derivative provides a formal approach to compute how a function changes at an infinitesimally small scale. A derivative of \( f(x) \) at \( x = c \) is defined as:

• \( f'(c) = \lim_{h \to 0} \frac{f(c + h) - f(c)}{h} \).

For our function \( f(x) = |x|^x \) at \( x = 0 \), we analyze this limit using \( f'(0) = \lim_{h \to 0} \frac{|h|^h - 1}{h} \). As \(|x|^x\) involves approximations turning non-linear near zero due to logarithmic behavior \( h \ln |h| \), the limit does not stabilize.
This indicates non-differentiability despite the graph's apparent smoothness due to its oscillatory and undefined nature close to zero, revealing intricate behaviors not captured by basic visual inspection.