Calculus I · Limits and Continuity · lesson
When a Limit Does Not Exist
Learn When a Limit Does Not Exist with plain-language explanations, guided examples, worked homework methods, interactive checks, and exam-style practice.
Where this chapter fits
Chapter 1: What a limit means
Build the neighborhood idea with motion, tables, holes, one-sided behavior, and graphs.
Reading lens: What are nearby outputs doing as the input approaches the target from both sides? Keep that question in view while reading When a Limit Does Not Exist; the worked mathematics is evidence for the idea, not a substitute for it.
This page connects One-Sided Limits From the Left and Right to How to Read Limits From a Graph. Read the explanation first, predict each example’s next move, and only then compare the written solution.
Learning objectives
Recognize and explain the major reasons a limit fails to exist: unequal one-sided limits, unbounded behavior, oscillation, or missing domain on one side.
How a Limit Can Fail
Writing is not a full explanation. A good solution identifies the behavior responsible.
Failure mode 1: the sides disagree
This is the jump behavior already seen:
Failure mode 2: unbounded behavior
For
as , the outputs become arbitrarily large and positive. As , they become arbitrarily negative. We write
The two-sided limit does not exist as a real number.
Failure mode 3: endless oscillation
For
the argument becomes arbitrarily large as . The sine function cycles between and infinitely often, so the outputs do not approach one number.
The function oscillates increasingly rapidly near zero and has no limit there.
Failure mode 4: the function exists only on one side
The function has no real values for . Therefore, the right-hand limit
exists, but an ordinary two-sided real limit at is not available because there is no left-side domain near zero.
At an endpoint of a domain or interval, a one-sided limit is often exactly the correct question. Later, continuity at endpoints will also use one-sided limits.
When asked why a limit does not exist, use one of these sentence frames:
• "The left-hand limit is and the right-hand limit is , and ." • "The function is unbounded near the target input." • "The function oscillates without approaching one output." • "The function has no domain values on one required side of the target."
Source & rights
Original instruction with traceable references.
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