Connecting Infinite Limits and Vertical Asymptotes
Use one-sided sign analysis to distinguish vertical asymptotes from removable holes.
College Board Topic 1.14 · CED p. 46 (PDF p. 51) · CED effective Fall 2020; project course outline 2026–27.
An infinite limit describes unbounded output near a finite input. Writing lim(x→c⁺)f(x)=+∞ means that sufficiently close inputs on the right produce outputs larger than any chosen positive threshold. Infinity is not a real function value.
A vertical asymptote x=c occurs when at least one one-sided limit is +∞ or −∞. Determine the sign on each side. For 1/(x−2), the denominator is negative on the left and positive on the right, so the limits are −∞ and +∞. For 1/(x−2)² both sides approach +∞.
A zero denominator alone does not prove a vertical asymptote: a common factor might cancel and leave a hole. Separate the two-sided finite-limit statement from an infinite-limit description. If the sides grow with opposite signs, do not write a single two-sided +∞ limit.
Let's work through it
- For 3/(x−1), approach 1 from below: denominator is negative and shrinks in magnitude. Reason: the quotient becomes arbitrarily negative.
- Approach 1 from above: denominator is positive and shrinks in magnitude. Reason: the quotient becomes arbitrarily positive.
- Thus x=1 is a vertical asymptote; left limit −∞, right limit +∞. Reason: unbounded one-sided behavior supplies the asymptote.
Try it from memory
Pause & retrieve
Try it before you peek.
Write your answer and the reason from memory. Then compare with the worked answer. Your self-assessment is a study decision, not an automatically graded score.
Question 1
What is lim(x→0)1/x²?
See every recall question with its answer
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Retrieval 1
What is lim(x→0)1/x²?
Worked answer and reason
+∞ as an infinite-limit description; there is no finite real limit.
The positive denominator shrinks to 0 from both sides.
Retrieval 2
Does (x²−1)/(x−1) have a vertical asymptote at 1?
Worked answer and reason
No; it has a removable hole.
For x≠1 it equals x+1 and has finite limit 2.
Catch the tempting mistake
- Infinity can be assigned as f(c) to repair continuity.
- Continuity here requires a finite real function value matching a finite limit.
Calculator workspace and limits
For calculus, set the angle selector to Radians. This scientific calculator evaluates arithmetic and functions. It does not graph arbitrary functions, find numerical roots, evaluate numerical derivatives, or compute definite integrals. Those tasks need an approved graphing tool; the May 2027 exam permits approved handheld graphing calculators and Bluebook’s built-in Desmos on calculator-required parts. A graph or decimal estimate helps you check a conclusion; your setup, hypotheses, units and justification still do the explaining.
Scientific calculator
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Put it to work
- Unit 1 problem 12 · No calculator · analytic
- Unit 1 problem 23 · No calculator · analytic
- Unit 1 problem 24 · No calculator · analytic
For the teacher
- Build a sign chart on either side of the target.
- Use the phrases “unbounded positive” and “unbounded negative” before the infinity notation.
Glossary and deeper reading · 4 refreshers
These editorial refreshers supplement the reading. Definitions and examples are added guidance; original source wording and recorded answers remain in their original context.
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Limit
A limit describes the value approached by outputs as inputs approach a specified target or grow without bound. For an input approaching a finite target, its value at that exact input does not determine the limit. For example, (x²−1)/(x−1) approaches 2 as x approaches 1 even though the original expression is undefined at 1. A finite two-sided limit requires the left and right approaches to agree.
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Function
A mathematical function assigns exactly one output to each input in its domain. Inputs can share an output, but one input cannot have two different outputs. For example, f(x)=x² assigns the output 9 to both 3 and −3. Function notation names the rule and its input; f(3) means the output when the input is 3.
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Vertical asymptote
A vertical asymptote x=c occurs when function values tend to positive or negative infinity as x approaches c from at least one side. Being unbounded near c is insufficient if the values oscillate without tending to either signed infinity. For example, 1/x has vertical asymptote x=0, tending to positive infinity from the right and negative infinity from the left. A zero denominator alone does not prove an asymptote because cancellation can instead leave a removable hole.
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One-sided limit
A one-sided limit uses inputs approaching a target from just one side. A left-hand limit uses smaller inputs, and a right-hand limit uses larger inputs. For example, a function that equals 0 left of 2 and 1 right of 2 has different one-sided limits at 2. The two-sided limit exists as a finite real number only when both one-sided limits exist and agree.