Calculus I · Limits and Continuity · lesson
Infinite Limits Explained
Learn Infinite Limits Explained with plain-language explanations, guided examples, worked homework methods, interactive checks, and exam-style practice.
Where this chapter fits
Chapter 4: Infinite behavior
Read vertical, horizontal, and slant asymptotes through sign analysis and dominant-term reasoning.
Reading lens: Is the function growing without bound near a finite input, or settling into end behavior as the input grows? Keep that question in view while reading Infinite Limits Explained; the worked mathematics is evidence for the idea, not a substitute for it.
This page connects Trigonometric Limit Decision Guide to Vertical Asymptotes and One-Sided Sign Analysis. Read the explanation first, predict each example’s next move, and only then compare the written solution.
Learning objectives
Interpret infinite-limit notation; determine one-sided signs near a vertical asymptote; distinguish unbounded behavior from an ordinary finite limit.
Infinite Behavior and Asymptotes
Infinite Limits
A finite limit asks whether outputs approach one real number. An infinite limit describes outputs whose magnitude grows without bound.
Infinite-Limit Notation
The statement
means that becomes arbitrarily large and positive as approaches .
The statement
means that becomes arbitrarily negative as approaches .
Infinity is not a real number. The notation records a direction of unbounded growth.
Imagine a thermometer with no top. Saying the reading approaches does not mean it arrives at a final number called infinity. It means that no matter how large a height you name, the reading eventually exceeds it near the target input.
\(1/x\) near zero
Consider .
From the right of zero, use small positive numbers:
| x | 1/x |
|---|---|
| 0.1 | 10 |
| 0.01 | 100 |
| 0.001 | 1000 |
Thus,
From the left, use small negative numbers:
| x | 1/x |
|---|---|
| -0.1 | -10 |
| -0.01 | -100 |
| -0.001 | -1000 |
Thus,
The one-sided behaviors differ, so the ordinary two-sided limit does not exist.
An odd power changes sign across the vertical asymptote; an even power remains positive on both sides.
Source & rights
Original instruction with traceable references.
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