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Directional limits
One-Sided Limits: Read Left and Right Separately
A one-sided limit restricts nearby inputs to one direction. This reveals jumps, vertical asymptotes, and domain boundaries that a two-sided statement can hide.
Direct answer
The function value at the target does not determine either one-sided limit.
When to use how to read one-sided limits
Choose the method from the top-level mathematical structure before manipulating symbols. Conditions are part of the task, not optional notes.
Ask first 1
Which structure controls the method?
What to do
Use one-sided limits near jumps, asymptotes, piecewise boundaries, and endpoints.
Ask first 2
Which condition can change the answer?
What to do
The function value at the target does not determine either one-sided limit.
Ask first 3
How should the result be checked?
What to do
Record the approach direction and compare the two directional results before stating a two-sided answer.
Compare the two approaches
Compute the left-hand limit using only x values below the target.
Compute the right-hand limit using only x values above the target.
A finite two-sided limit exists exactly when both directional limits exist and are equal.
If the signs or finite values differ, report that the two-sided limit does not exist and retain both directional results.
Worked examples
Positive inputs produce positive unbounded values.
Negative inputs produce negative unbounded values.
The left limit is −1 and the right limit is 1.
Common mistakes
- • Checking only one direction.
- • Using the point value as the limit.
- • Writing infinity without its sign.
- • Averaging unequal one-sided limits.
Error clinic: locate the first invalid step
Incorrect attempt
Why it fails
The two directions have opposite signs.
Correction
Incorrect attempt
Why it fails
The function value or an average cannot replace directional behavior.
Correction
Practice the same idea with a changed structure
Name the rule and its conditions before revealing the reference answer. Explain every sign and factor.
Check before solving
Negative inputs produce negative unbounded values.
Reveal the reference answer
Reference answer
Check before solving
The left limit is −1 and the right limit is 1.
Reveal the reference answer
Reference answer
Try the worked examples in the calculator
Each link opens the matching calculator with the expression, variable and required conditions already filled in.
Open the calculator without a preset: 極限Frequently asked questions
Does the function need to be defined at the target?+
No. Limits depend on nearby values, not necessarily the point value.
Can both sides be infinite and still disagree?+
Yes. +∞ and −∞ are different directional behaviors, so no common two-sided limit exists.
Why specify left or right at a domain endpoint?+
Only one direction may contain valid domain points.
Conclusion
Evaluate both directions explicitly, preserve their signs, and declare a two-sided limit only when the directional results match.
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