Calculus I · Limits and Continuity · lesson
Function Value vs. Limit
Learn Function Value vs. Limit 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 Function Value vs. Limit; the worked mathematics is evidence for the idea, not a substitute for it.
This page connects Limits at Holes and Undefined Points to One-Sided Limits From the Left and Right. Read the explanation first, predict each example’s next move, and only then compare the written solution.
Learning objectives
Determine and separately; explain why changing one isolated function value does not change the limit.
The Function Value and the Limit Are Different Questions
The expression asks:
What output has the function assigned at the single input ?
The expression
asks:
What value do nearby outputs approach when nearby inputs approach ?
These answers can agree, disagree, or one may fail to exist.
The dot and the road
Imagine a road drawn toward the point , but someone places a separate dot at . The road tells you where nearby points are going. The separate dot tells you the actual value at .
Thus,
The limit follows the road, not the misplaced dot.
limit-hole-01Evaluate .
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Factor the difference of squares.
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A piecewise value does not control the limit
Let
Find and .
Show worked solution
The piecewise definition directly states
For all nearby inputs , the rule is . Therefore,
So
The nearby parabola determines the limit; the filled point determines the function value.
A single point has no control over a two-sided limit. You may change , delete it, or define it later, and the limit remains the same as long as all nearby values with remain unchanged.
Suppose for every , while . Find
Answer. , but
Source & rights
Original instruction with traceable references.
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