What a One-Sided Limit Means
When we write a limit such as , we are asking what value approaches as gets close to from both sides.
But sometimes we only want to approach the input from one direction. That's where one-sided limits come in.
A left-hand limit asks what happens as approaches using values less than . A right-hand limit asks what happens using values greater than .
The superscript means "from the left," while means "from the right."
Why One-Sided Limits Matter
One-sided limits are especially important when a function behaves differently on opposite sides of a point. This happens frequently with piecewise functions, jump discontinuities, and vertical asymptotes.
For example, imagine that a graph approaches 3 from the left but approaches 7 from the right as . There is no single value that the function approaches from both sides.
Because the two one-sided limits are different, the ordinary two-sided limit does not exist.
The Key Rule for Two-Sided Limits
A two-sided limit exists only when the left-hand and right-hand limits both exist and are equal.
This is one of the most important limit facts to remember for AP Calculus. If the two sides disagree, you cannot simply average them, choose one, or substitute the function's actual value.
How to Evaluate One-Sided Limits
- Identify the direction. Look for or .
- Approach the target from that side only. For a left-hand limit, use values smaller than the target. For a right-hand limit, use values larger than the target.
- Use the appropriate branch if the function is piecewise.
- Determine the value being approached.
- Compare the two sides if you are asked for the ordinary two-sided limit.
Worked Examples
Find .
Find .
Suppose
Find the left-hand and right-hand limits as .
Determine .
This is an infinite limit. It does not mean that the limit equals an ordinary real number. Instead, it describes the function's unbounded behavior near the point.
One-Sided Limits and Vertical Asymptotes
One-sided limits are essential when identifying vertical asymptotes. If a function approaches positive or negative infinity as the input approaches a particular value, the corresponding vertical line is a vertical asymptote.
For example, consider . Near , the denominator approaches zero, and the sign of the denominator depends on which side we approach from.
Since the function behaves differently from the two sides, checking one-sided limits tells us exactly what happens on each side of the vertical asymptote.
One-Sided Limits and Piecewise Functions
Piecewise functions are one of the most common AP Calculus settings for one-sided limits. The key is to choose the correct formula based on the direction of approach.
If a piecewise function changes formulas at , then:
- For , use the branch defined for values less than .
- For , use the branch defined for values greater than .
This is why the direction symbol matters so much. The same function can have two completely different one-sided limits at the same input.
One-Sided Limits and Continuity
Continuity at a point requires the ordinary two-sided limit to exist and agree with the function's value. Therefore, both one-sided limits must agree as well.
So, to check continuity at , you can think through the following chain:
- Does the function exist at ?
- Does the left-hand limit exist?
- Does the right-hand limit exist?
- Are the two one-sided limits equal?
- Does their common value equal ?
If either one-sided limit disagrees with the other, the two-sided limit does not exist, and the function cannot be continuous at that point.
Reading One-Sided Limits from a Graph
On an AP Calculus graph question, you usually do not need an algebraic calculation. Instead, visually track the graph as approaches the target.
From the left
Imagine moving toward the target from the left side of the graph. Ignore what happens on the right. Ask: "What -value does the graph approach?"
From the right
Imagine moving toward the target from the right side of the graph. Again, ignore the opposite side and identify the -value being approached.
The actual value of the function at the point may be irrelevant. An open circle, filled dot, or even a missing point does not automatically determine the limit. The limit is about what the graph approaches, not necessarily where the function is defined.
Common Mistakes
Remember: means approach from values less than the target, while means approach from values greater than the target.
For a left-hand limit, use the formula that applies immediately to the left of the target. For a right-hand limit, use the formula that applies immediately to the right. The direction of approach determines which branch matters.
Never calculate only one side and assume the other side matches. If the problem involves a possible jump, piecewise definition, or graph with different behavior on either side, explicitly compare the left-hand and right-hand limits.
The value tells you where the function is actually defined. The limit asks where the function values are heading as the input approaches the target. These can be different.
Writing describes unbounded behavior. Infinity is not a real number that the function reaches; it tells you that the function grows without bound as the input approaches the target from the specified side.
Practice Problems
Find .
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Find .
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Suppose
Find the left-hand and right-hand limits as .
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Determine .
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Suppose and . Does exist?
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