C03 · Unit C1 Limits and continuity
Squeeze reasoning and asymptotes
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- Time
- About 45 minutes, untimed
- Course framework reference
- AP 1.8–1.9, 1.14–1.15
- Builds on
- Part of
Progress on this lesson counts in every pathway listed.
Understand the idea
The squeeze theorem traps a function between two functions with the same limit. Because |sin u| is less than or equal to 1, minus x squared is less than or equal to x squared sin(1 over x) is less than or equal to x squared, giving limit zero at x equals 0. Infinite limits describe unbounded one-sided behavior; they are not finite values. For rational functions at infinity, divide by the largest denominator power and track leading terms and signs.
Worked example
For f(x) equals (3x squared +1)/(2x squared minus 5), division by x squared gives (3+1 over x squared)/(2 minus 5 over x squared) tends to 3 over 2 as x tends to ∞. Thus y equals 3 over 2 is a horizontal asymptote. Horizontal asymptotes may be crossed at finite inputs.
Explore the mathematics
Squeeze reasoning and asymptotes
Bound an oscillating function between two curves with the same limit.
Scroll sideways to see the full diagram.
Predict before continuing
The middle function oscillates. Can its limit still exist?
Read the complete explanation
- The middle function oscillates. Can its limit still exist?
- Bound it above and below using −1≤sin(1/x)≤1.
- Both bounding functions approach 0 as x→0.
- The squeeze theorem gives limit 0. At |x|≤0.25, the magnitude is at most 0.0625.
On the positive branch, (3x²+1)/(2x²−5) approaches the horizontal line y=3/2 as x increases. The graph shows x≥2 only; the original domain also excludes x=±√(5/2).
Discuss: explain what each label represents, then draw a related example with different values.
A common misconception
An infinite limit does not mean the function is defined with value infinity. At 1 over x near zero the sides diverge in opposite directions.
Investigate and explain
Zoom toward zero on xsin(1 over x), using its envelope plus or minus |x|. Explain how the envelope settles a question that the oscillating graph alone makes difficult.
Identify one step in the worked example, name the mathematical fact that permits it, and state its domain conditions.