Calculus I · Limits and Continuity · lesson
Polynomial and Slant Asymptotes
Learn Polynomial and Slant Asymptotes 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 Polynomial and Slant Asymptotes; the worked mathematics is evidence for the idea, not a substitute for it.
This page connects Limits at Infinity and Horizontal Asymptotes to Radical Limits at Infinity. Read the explanation first, predict each example’s next move, and only then compare the written solution.
Learning objectives
Use polynomial division when the numerator degree exceeds the denominator degree; identify a slant asymptote when the degree difference is one.
Polynomial and Slant Asymptotes
When the numerator degree is exactly one larger than the denominator degree, long division produces
The remainder term approaches zero, leaving a slant or oblique asymptote.
Slant asymptote by division
Find the end behavior of
Show worked solution
Divide by :
As ,
Therefore the graph approaches
This is a slant asymptote. The function does not approach a finite number, but its difference from approaches zero:
Source & rights
Original instruction with traceable references.
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