One-sided limit
A limit where x approaches a from only one side: from the left (x below a) or from the right (x above a). The ordinary limit exists only if both agree.
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Analogy
Two people walking to the same bridge from opposite banks. They meet only if both reach the same spot.
Example
For x > 0: f(x) = x/x = 1.For x < 0: f(x) = -x/x = -1.Right-hand limit is 1, left-hand limit is -1.They differ, so no limit at 0.
In practice
Needed for jumps, absolute values and piecewise functions, where the two sides behave differently.
Practice
Flashcards
- When does a two-sided limit exist in terms of one-sided limits?When the left-hand and right-hand limits exist and are equal.
Questions people ask
How do I write a one-sided limit?
Put a small plus or minus after the number. Write x approaching a from the right as x tends to a plus, and from the left as x tends to a minus. The two-sided limit exists exactly when both one-sided limits exist and are equal to each other.
When do I need one-sided limits?
Use them for piecewise functions, absolute values, floor functions and square roots near the edge of their domain. For example, the square root of x only has a right-hand limit at 0, since negative inputs are not allowed in the real numbers.