Infinite Limits and Asymptotes

Learn limits at infinity, vertical asymptotes.

Lesson 3 of 32 parts Practice Flashcards

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Limits at Infinity

Objective: Explain limits at infinity, apply the stated method, and check its conditions.

A limit at infinity describes output behavior for arbitrarily large input magnitude. For rational functions with equal numerator and denominator degrees, the limit is the leading-coefficient ratio. Divide by the highest input power to justify discarding lower-order terms. A horizontal asymptote describes end behavior and may be crossed.

Formulas

$limₓ→∞(ax+b)/(cx+d)=a/c, c≠0$

Worked examples

Try itFind limₓ→+∞ (6x+7)/(7x+1).Show answer

Step 1: Divide both numerator and denominator by x. Step 2: The ratio becomes (6+7/x)/(7+1/x). Step 3: As 1/x→0, the limit is 6/7.

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Limits at Infinity

Objective: Explain limits at infinity, apply the stated method, and check its conditions.

Formulas

$limₓ→∞(ax+b)/(cx+d)=a/c, c≠0$

Worked examples

Try itFind limₓ→+∞ (6x+7)/(7x+1).Show answer

Step 1: Divide both numerator and denominator by x. Step 2: The ratio becomes (6+7/x)/(7+1/x). Step 3: As 1/x→0, the limit is 6/7.

Vertical Asymptotes

Objective: Explain vertical asymptotes, apply the stated method, and check its conditions.

A vertical asymptote x=a occurs when at least one one-sided function limit is unbounded as x approaches a. A zero denominator is only a candidate; factor cancellation may instead produce a hole. Check numerator behavior and both approach directions. Not every zero denominator produces a vertical asymptote.

Formulas

For A/(x−h), A≠0: vertical asymptote x=h

Worked examples

Try itFor f(x)=6/(x−7), find the x-value of its vertical asymptote.Show answer

Step 1: The denominator vanishes at the asymptote. Step 2: Solve x−7=0; the numerator is nonzero. Step 3: The vertical asymptote is x=7.

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