Calculus I · Limits and Continuity · lesson

Polynomial and Slant Asymptotes

Concept

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

f(x)=\text{linear quotient}+\frac{\text{remainder}}{q(x)}.

The remainder term approaches zero, leaving a slant or oblique asymptote.

Worked example

Slant asymptote by division

Find the end behavior of

f(x)=\frac{x^2+1}{x-1}.

Show worked solution

Divide $x^2+1$ by $x-1$ :

\frac{x^2+1}{x-1}=x+1+\frac2{x-1}.

As $x\to\pm\infty$ ,

\frac2{x-1}\to0.

Therefore the graph approaches

\boxed{y=x+1}.

This is a slant asymptote. The function does not approach a finite number, but its difference from $x+1$ approaches zero:

\lim_{x\to\pm\infty}[f(x)-(x+1)]=0.

After the explanation

Use the section idea

Reading lens

Is the function growing without bound near a finite input, or settling into end behavior as the input grows?

Mental model

Vertical asymptotes describe local blow-up near an excluded finite input; end-behavior asymptotes describe the long-run trend as inputs grow in magnitude.

Decision

Near a denominator zero, build a sign chart for each side; at infinity, compare dominant powers or divide to expose the lasting term.

Common trap

Do not merge positive and negative infinity, and remember that square roots produce absolute values when factoring a large squared input.

Check yourself

You understand the section when you can predict signs and asymptotes before doing detailed algebra, then verify them with the expression.

Source & rights

Original instruction with traceable references.

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