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Magazine Chapter 2: Problem 19
\(\lim _{x \rightarrow 0} \frac{|x|}{x}\) is (A) 0 (B) nonexistent (C) 1 (D) -1
Short Answer
Expert verified
The limit does not exist because the one-sided limits are different.
Step by step solution
01
Understanding the Limit
We start with understanding the expression \(\lim _{x \rightarrow 0} \frac{|x|}{x}\). This is a limit problem, and the expression involves the absolute value of \(x\), \(|x|\). The absolute value function behaves differently for positive and negative values of \(x\), which causes us to consider one-sided limits.
02
Evaluating the Right-Hand Limit (as \(x \to 0^+\))
For \(x > 0\), \(|x| = x\). So the expression becomes \(\frac{x}{x} = 1\). Therefore, the right-hand limit as \(x\) approaches 0 is 1. Mathematically, this means: \(\lim _{x \rightarrow 0^+} \frac{|x|}{x} = 1\).
03
Evaluating the Left-Hand Limit (as \(x \to 0^-\))
For \(x < 0\), \(|x| = -x\). Thus, the expression turns into \(\frac{-x}{x} = -1\). So, the left-hand limit as \(x\) approaches 0 is -1. In mathematical terms, \(\lim _{x \rightarrow 0^-} \frac{|x|}{x} = -1\).
04
Comparing One-Sided Limits
The right-hand limit and the left-hand limit as \(x\) approaches 0 are different. Specifically, \(\lim _{x \rightarrow 0^+} \frac{|x|}{x} = 1\) and \(\lim _{x \rightarrow 0^-} \frac{|x|}{x} = -1\). Since these limits are not equal, the overall limit does not exist.
05
Final Conclusion
The overall limit \(\lim _{x \rightarrow 0} \frac{|x|}{x}\) does not exist because the one-sided limits are not equal. Therefore, the correct choice is (B) nonexistent.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
One-Sided Limits
In the world of calculus, one-sided limits explore the behavior of functions as they approach a specific point from one direction鈥攅ither from the left or from the right. This is crucial when evaluating limits involving functions that behave differently on either side of a point, particularly for non-continuous functions.
- Right-Hand Limit (RHL): This limit considers approaching the target value from the right-hand side, denoting positive direction. It is symbolized by \(x \rightarrow a^+\).
- Left-Hand Limit (LHL): Likewise, this limit considers approaching from the left-hand side, or negative direction, denoted as \(x \rightarrow a^-\).
Absolute Value Function
The absolute value function, \(|x|\), is a core mathematical concept defined as the non-negative value of \(x\). It effectively "strips" any negative sign from the number, so \( |x| = x\) if \(x \geq 0\) and \( |x| = -x\) if \(x < 0\).
- For positive numbers or zero, the function is straightforward: the absolute value is the number itself.
- For negative numbers, the absolute value is the positive counterpart of that number.
Limit Existence
The existence of a limit at a point is heavily reliant on the behavior of one-sided limits. A limit of a function \(f(x)\) as \(x\) approaches \(a\) exists if and only if both the right-hand limit and the left-hand limit are equal.
- If \(\lim_{x \to a^-} f(x) = \lim_{x \to a^+} f(x)\), the overall limit exists and equals that common value.
- If the one-sided limits differ, the limit at \(a\) does not exist. This mismatch generally points to a break or jump in the function, reflecting some discontinuity at \(a\).
Right-Hand Limit
A right-hand limit is examined when we want to understand how a function behaves as its input variable approaches a specific number from the right. Symbolized as \(x \to a^+\), it's particularly important when a function displays different expressions on either side of a point.
When dealing with expressions involving absolute values, such as \(\lim _{x \rightarrow 0^+} \frac{|x|}{x}\), the expression simplifies according to the positive case. For example:
When dealing with expressions involving absolute values, such as \(\lim _{x \rightarrow 0^+} \frac{|x|}{x}\), the expression simplifies according to the positive case. For example:
- If \(x > 0\), \( |x| = x\), turning the function into \(\frac{x}{x} = 1\).
Left-Hand Limit
Evaluating the left-hand limit determines the function's behavior as its input approaches a target value from the left, marked as \(x \to a^-\). When functions behave differently on either side of a point鈥攃ommon with absolute values鈥攖hese limits provide essential insights.
Taking the scenario of \(\lim _{x \rightarrow 0^-} \frac{|x|}{x}\):
Taking the scenario of \(\lim _{x \rightarrow 0^-} \frac{|x|}{x}\):
- If \(x < 0\), the expression \( |x| = -x\) adjusts the function to \(\frac{-x}{x} = -1\).
