Why Absolute Value Makes Limits Different
Absolute value functions are continuous, but they behave differently on the two sides of the point where the expression inside the absolute value changes sign. That makes them especially important when evaluating limits.
The key definition is:
So an expression involving absolute value may actually represent two different algebraic expressions, depending on whether the quantity inside the absolute value is positive or negative.
For example, is equal to when , but it is equal to when .
The Most Important Question: What Happens Near the Point?
When evaluating a limit such as , the first thing to determine is whether the expression inside the absolute value changes sign at the point you are approaching.
If it does not change sign, you can usually remove the absolute value by determining its sign near the point. If it does change sign, you must analyze the two sides separately.
This is why one-sided limits are so useful for absolute value problems.
A Step-by-Step Method
- Identify the expression inside the absolute value. Call it something like .
- Find where that expression is zero. Those points are where the absolute value can switch from one algebraic form to another.
- Determine the sign on each side of the point.
- Rewrite the absolute value piecewise if necessary.
- Evaluate the left-hand and right-hand limits.
- Compare the two answers. If they are equal, the two-sided limit exists. If they differ, the limit does not exist.
Worked Examples
Evaluate .
Evaluate .
Evaluate .
Evaluate .
When You Can Substitute Directly
It is important not to overcomplicate every absolute value limit. If the expression is continuous at the point and the denominator is not zero, direct substitution works exactly as it does for ordinary functions.
For example, consider . The function inside the absolute value is continuous, and the absolute value function itself is continuous. Therefore, the composition is continuous at .
The important distinction is that absolute value itself does not cause a limit to fail. Problems arise when the expression inside the absolute value changes sign, particularly when it appears in a denominator or when you need to compare one-sided behavior.
Absolute Value and One-Sided Limits
A very common AP Calculus pattern is an expression such as . The absolute value behaves differently depending on whether approaches the point from the left or right.
For , we have . Therefore:
This technique is worth memorizing because it appears repeatedly in limits involving absolute values, piecewise functions, and rational expressions.
Common Mistakes
The identity is true only when . For negative values of , .
If the expression inside the absolute value changes sign at the point of interest, you cannot automatically assume the left- and right-hand behavior is the same. Check both one-sided limits.
Near , the sign of depends on which side of you are approaching from. In particular, on the left and on the right.
A function can have a sharp corner and still have a perfectly well-defined limit. For example, has a corner at 0, but exists. A corner is primarily a differentiability issue, not automatically a limit issue.
Practice Problems
Evaluate .
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Evaluate .
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Evaluate .
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Evaluate and .
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