When dealing with limits involving absolute value functions, the key is to understand the piecewise nature of the absolute value. The absolute value of a function, |f(x)|, is defined differently depending on whether f(x) is non-negative or negative. This often requires examining left-hand and right-hand limits separately.
Understanding Absolute Value
The absolute value function is defined as:
|f(x)| =
- f(x), if f(x) ≥ 0
- -f(x), if f(x) < 0
Rules for Evaluating Absolute Value Limits
The general strategy for evaluating limits of absolute value functions involves these steps:
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Consider the Piecewise Definition: Rewrite the absolute value function as a piecewise function, based on where the expression inside the absolute value is positive, negative, or zero.
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Evaluate Left-Hand and Right-Hand Limits: Calculate the left-hand limit (approaching from the left, x → a-) and the right-hand limit (approaching from the right, x → a+) separately. This is crucial because the function might behave differently on either side of the point in question.
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Compare the Limits:
- If the left-hand limit and the right-hand limit are equal, then the limit exists and is equal to that common value.
- If the left-hand limit and the right-hand limit are not equal, then the limit does not exist.
Examples and Practical Insights
Here are a few examples to illustrate the process:
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Example 1: Find limx→0 |x|/x.
- For x > 0, |x| = x, so |x|/x = x/x = 1.
- For x < 0, |x| = -x, so |x|/x = -x/x = -1.
- Therefore, limx→0+ |x|/x = 1 and limx→0- |x|/x = -1.
- Since the left-hand and right-hand limits are not equal, the limit does not exist.
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Example 2: Find limx→2 |x - 2|.
- For x > 2, |x - 2| = x - 2.
- For x < 2, |x - 2| = -(x - 2) = 2 - x.
- limx→2+ |x - 2| = limx→2+ (x - 2) = 0.
- limx→2- |x - 2| = limx→2- (2 - x) = 0.
- Since both limits are 0, limx→2 |x - 2| = 0.
Summary Table
| Condition | Rule |
|---|---|
| f(x) ≥ 0 | |
| f(x) < 0 | |
| limx→a+ | f(x) |
| limx→a+ | f(x) |
Key Takeaways
- Always consider the piecewise definition of the absolute value.
- Calculate left-hand and right-hand limits separately.
- The limit exists only if the left-hand and right-hand limits are equal.
- Studying the function to the left and right of x=a helps understand the limit as x→a.
