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Magazine Chapter 3: Problem 44
For the following exercises, use the limit definition of derivative to show that the derivative does not exist at \(x=a\) for each of the given functions. $$ f(x)=\frac{|x|}{x}, x=0 $$
Short Answer
Expert verified
The derivative of \(f(x) = \frac{|x|}{x}\) at \(x=0\) does not exist because the left-hand and right-hand limits are infinite and do not agree.
Step by step solution
01
Recall the Limit Definition of Derivative
The derivative of a function \(f(x)\) at the point \(x=a\) is defined as \(f'(a) = \lim_{h \to 0} \frac{f(a + h) - f(a)}{h}\). Our goal is to determine whether this limit exists when \(a=0\).
02
Substitute and Simplify the Function
Substitute \(f(x) = \frac{|x|}{x}\) into the derivative definition at \(x=0\). We have \(f'(0) = \lim_{h \to 0} \frac{f(h) - f(0)}{h}\). Note that \(f(0) = \frac{|0|}{0}\) is undefined, so instead evaluate the limit directly.
03
Determine \(f(h)\) for Positive and Negative \(h\)
For \(h > 0\), \(f(h) = \frac{|h|}{h} = 1\). For \(h < 0\), \(f(h) = \frac{|h|}{h} = -1\). Evaluate the limit separately for \(h > 0\) and \(h < 0\) to check for the existence of the limit.
04
Evaluate the Right-Hand Limit
Calculate \(\lim_{h \to 0^+} \frac{f(h) - f(0)}{h} = \lim_{h \to 0^+} \frac{1 - 0}{h} = \lim_{h \to 0^+} \frac{1}{h} = +\infty\). The right-hand limit does not exist as it approaches infinity.
05
Evaluate the Left-Hand Limit
Calculate \(\lim_{h \to 0^-} \frac{f(h) - f(0)}{h} = \lim_{h \to 0^-} \frac{-1 - 0}{h} = \lim_{h \to 0^-} \frac{-1}{h} = -\infty\). The left-hand limit does not exist as it approaches negative infinity.
06
Conclusion on the Existence of Derivative
Since the right-hand limit is \(+\infty\) and the left-hand limit is \(-\infty\), \(\lim_{h \to 0} \frac{f(h) - f(0)}{h}\) does not exist. Therefore, the derivative of \(f(x)\) at \(x=0\) does not exist.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Limit definition of derivative
The limit definition of derivative is a foundational concept in calculus. It allows us to determine the derivative, which represents the rate of change of a function at a specific point. To calculate the derivative of a function \( f(x) \) at a point \( x = a \), we use the formula: \[ f'(a) = \lim_{h \to 0} \frac{f(a + h) - f(a)}{h}\]This expression calculates the ratio of the change in the function鈥檚 value to the change in \( x \) as \( h \) approaches zero. If this limit exists, it gives us the slope of the tangent line to the function at \( x = a \).
- If the limit is finite, \( f'(a) \) exists, indicating a defined slope at that point.
- If the limit does not exist, the function does not have a derivative at that point.
Right-hand limit
The right-hand limit is a specific type of limit that focuses on values approaching from one side. In calculus, it is used to examine how a function behaves as the variable approaches a specified point from the right side. Mathematically, it is denoted as \( \lim_{h \to 0^+} f(h) \). Considering our function example, as \( h \) approaches zero from the positive side, we substitute \( f(h) = \frac{|h|}{h} = 1 \) because \( |h| = h \) for positive \( h \).
- Evaluating the limit as \( h \to 0^+ \) gives \( \lim_{h \to 0^+} \frac{1}{h} = +\infty \).
- This result indicates that the slope tends to infinity, suggesting no finite or defined value for the derivative at this point when approaching from the right.
Left-hand limit
Understanding the left-hand limit is crucial in determining the behavior of a function as it approaches a specific point from the left side. In mathematics, it can be represented as \( \lim_{h \to 0^-} f(h) \). For our function \( \frac{|x|}{x} \), when \( h \) is negative and approaching zero, \( |h| = -h \) which makes \( f(h) = -1 \).
- Calculating the left-hand limit yields \( \lim_{h \to 0^-} \frac{-1}{h} = -\infty \).
- The result of \(-\infty\) implies that the derivative's slope is unbounded and undefined, just like with the right-hand limit.
Infinity in limits
When discussing limits, encountering infinity typically means the function is unbounded in that direction. In our exercise, both the right-hand and left-hand limits result in \( +\infty \) and \( -\infty \), respectively. This divergent behavior occurs because:
- The function value divides a constant by an increasingly small number \( h \), pushing the result toward infinity.
- Since the function's limits differ on either side of the approaching value, we cannot assign a single, finite derivative at that point.
