Calculus I · Limits and Continuity · lesson
Infinity Minus Infinity Limits
Learn Infinity Minus Infinity Limits 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 Infinity Minus Infinity Limits; the worked mathematics is evidence for the idea, not a substitute for it.
This page connects Radical Limits at Infinity to Complete Rational Function Asymptote Analysis. Read the explanation first, predict each example’s next move, and only then compare the written solution.
The form
A difference of two large expressions is indeterminate. Rationalization can reveal the finite remainder.
Rationalize at infinity
Evaluate
Show worked solution
Multiply by the conjugate:
Since for large positive , divide numerator and denominator by :
Now take the limit:
Exam-level: negative infinity changes the conjugate calculation
Evaluate
Show worked solution
Rationalize:
For , factor from the radical:
Then the denominator becomes
Therefore,
Taking the limit gives
Source & rights
Original instruction with traceable references.
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