AP® Calculus AB review sheet from Aim for Five (aimforfive.com/calc-ab/units/1/1-14)
Unit 1 · Topic 1.14
1.14 Connecting Infinite Limits and Vertical Asymptotes
When a function's outputs grow without bound as x approaches c, you describe that with an infinite limit, written ∞ or −∞. The line x = c is then a vertical asymptote. The sign of the function on each side tells you whether the graph shoots up or down.
Key terms
- infinite limit
- unbounded
- vertical asymptote
- one-sided behavior
Infinite limits
lim (x→c⁺) f(x) = ∞ means the outputs become larger than any number you pick as x approaches c from the right. Similarly, −∞ means they become more and more negative.
An infinite limit is not a real number, so technically the limit does not exist. Writing ∞ or −∞ is more informative: it tells you how the limit fails. If the right side goes to ∞ and the left side to −∞, the two-sided limit doesn't exist even in that sense.
Vertical asymptotes
The line x = c is a vertical asymptote of f if at least one one-sided limit at c is ∞ or −∞. You only need one side.
Typical sources: a rational function whose denominator is 0 at c but whose top is not (after canceling); ln x at x = 0 (lim (x→0⁺) ln x = −∞); and tan x and sec x at odd multiples of π/2.
Finding the sign on each side
When substitution gives (nonzero number)/0, the quotient is unbounded. To find the sign, test a number just to the side. For example, use x = 3.01 to see the sign just right of 3.
Or reason with factors: decide whether each factor is positive or negative just to that side, then multiply the signs. A squared factor like (x − 2)² is positive on both sides, so both one-sided limits have the same sign.
| Top near c | Bottom near c | Quotient heads to |
|---|---|---|
| Positive | 0 through positive values | ∞ |
| Positive | 0 through negative values | −∞ |
| Negative | 0 through positive values | −∞ |
| Negative | 0 through negative values | ∞ |
Working with infinite limits
∞ is not a number, so the usual limit properties don't apply directly. Some patterns are safe: if f(x)→∞ and g(x)→5, then f(x) + g(x)→∞ and f(x)·g(x)→∞ (since 5 > 0). Others are undecided: ∞ − ∞, 0·∞ and ∞/∞ can come out to anything, so you'd need to rewrite.
Graphs help too. Near a vertical asymptote, a curve that climbs on both sides, like 1/x² near 0, has lim (x→0) 1/x² = ∞ from both sides. A curve that climbs on one side and falls on the other, like 1/x near 0, has different one-sided infinite limits.
Worked examples
Try each one yourself first, then open the solution.
- Example 1
Signs on both sides of two asymptotes
Let f(x) = (x + 2) / (x² − 9). Find the one-sided limits at x = 3 and at x = −3.
Show the solutionHide the solution
- Step 1: Write the bottom as (x − 3)(x + 3). At x = 3 the top is 5 and the bottom is 0, so both one-sided limits are infinite.
- Step 2: Just right of 3 (like 3.01): x − 3 > 0 and x + 3 > 0, so the bottom is a tiny positive number. 5 over tiny positive → ∞.
- Step 3: Just left of 3 (like 2.99): x − 3 < 0 and x + 3 > 0, so the bottom is tiny negative. 5 over tiny negative → −∞.
- Step 4: At x = −3 the top is −1. Just right of −3 (like −2.99): x − 3 < 0 and x + 3 > 0, so the bottom is tiny negative. −1 over tiny negative → ∞.
- Step 5: Just left of −3 (like −3.01): x − 3 < 0 and x + 3 < 0, so the bottom is tiny positive. −1 over tiny positive → −∞.
Answer: lim (x→3⁺) f(x) = ∞, lim (x→3⁻) f(x) = −∞, lim (x→−3⁺) f(x) = ∞, lim (x→−3⁻) f(x) = −∞. Both x = 3 and x = −3 are vertical asymptotes.
- Example 2
Trap: a zero denominator that is not an asymptote
Does g(x) = (x² − 4) / (x − 2) have a vertical asymptote at x = 2?
Show the solutionHide the solution
- Step 1: Substitution gives 0/0, not (nonzero)/0. That's the sign to factor.
- Step 2: (x² − 4) / (x − 2) = (x − 2)(x + 2) / (x − 2) = x + 2 for x ≠ 2.
- Step 3: lim (x→2) g(x) = 4, a finite number.
Answer: No. g has a hole at (2, 4), not a vertical asymptote, because the limit at 2 is 4.
Common mistakes
- Saying a function has a vertical asymptote wherever the denominator is 0, without checking whether the factor cancels.
- Guessing the sign of an infinite limit. Test a value just to that side or track the sign of each factor.
- Writing lim (x→3) f(x) = ∞ when the two sides go in opposite directions. Then you must write the one-sided limits separately.
On the exam
- Multiple-choice questions ask for vertical asymptotes of a rational function or for the value of a one-sided infinite limit.
- When a question asks you to justify a vertical asymptote, give a one-sided limit that equals ∞ or −∞.
Connected topics
Videos
Check yourself
4 questions on 1.14 Connecting Infinite Limits and Vertical Asymptotes. Pick an answer to see if you got it, and why.
What is lim (x→2⁻) (x + 1)/(x² − 4) ?
What is lim (x→1⁺) (x − 3)/ln x ?
When x = 1 is substituted into (x² + 3)/(x − 1), the result is 4/0. Which of the following is true about lim (x→1) (x² + 3)/(x − 1) ?
Which of the following is the correct notation for the statement "as x approaches 3 from the left, f(x) decreases without bound"?
0 of 4 answered