Calculus I · Limits and Continuity · review

Infinite Limits and Asymptotes Review

Section 4 Summary

Summary

• Infinite limits describe unbounded behavior; infinity is not a real number. • Remaining denominator zeros after cancellation give vertical asymptote candidates. • Sign charts determine $+\infty$ versus $-\infty$ on each side. • Rational end behavior is controlled by degrees and leading coefficients. • If the degree difference is one, polynomial division may reveal a slant asymptote. • At negative infinity, never replace $\sqrt{x^2}$ by $x$ ; use $|x|=-x$ .

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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