Consider f(x) = 1/(x−3). Which of the following best describes the behavior of f near x = 3?
Alim(x→3) f(x) = ∞, because both one-sided limits go to +∞
Blim(x→3) f(x) does not exist; the left-side limit is −∞ and the right-side limit is +∞
Clim(x→3) f(x) = 0, because the denominator becomes zero
Dlim(x→3) f(x) = ∞ or −∞ depending on which side you approach from, so the limit equals ±∞
From the right, (x−3) is small and positive, so 1/(x−3) → +∞. From the left, (x−3) is small and negative, so 1/(x−3) → −∞. Because the one-sided limits disagree, the two-sided limit does not exist — not even as ±∞. Option A is wrong because the left side goes to −∞, not +∞. Option C reverses the idea: a zero denominator causes the function to blow up, not approach zero. Option D misuses the notation ±∞ as if it were a single value.
Question 2 Multiple Choice
A function has a horizontal asymptote at y = 2. Which concept does this correspond to?
AAn infinite limit: lim(x→a) f(x) = ∞ for some finite a
BA limit at infinity: lim(x→∞) f(x) = 2
CAn infinite limit from only one side near x = 2
DA vertical asymptote where the output is bounded by 2
A horizontal asymptote describes the output settling toward a finite value as the input grows without bound — that is a limit at infinity, lim(x→∞) f(x) = 2. An infinite limit is the opposite scenario: the input approaches a finite value (x → a) while the output blows up. These two concepts are easy to conflate because both involve the word 'infinite,' but one refers to an infinite output near a finite input (vertical asymptote), and the other refers to a finite output as the input becomes infinite (horizontal asymptote).
Question 3 True / False
Writing lim(x→a) f(x) = ∞ means the limit exists and equals the number infinity.
TTrue
FFalse
Answer: False
Infinity is not a real number, so a limit that 'equals' ∞ technically does not exist in the standard real-number sense. The notation lim(x→a) f(x) = ∞ is a precise way of reporting a specific kind of non-existence: the function grows without bound as x approaches a. It conveys useful directional information about the failure mode, but it does not mean the limit has a real-number value.
Question 4 True / False
If lim(x→a⁺) f(x) = +∞ and lim(x→a⁻) f(x) = −∞, then lim(x→a) f(x) = ∞.
TTrue
FFalse
Answer: False
For the two-sided limit to exist (even as ∞), both one-sided limits must agree. Here the right-side limit is +∞ and the left-side limit is −∞ — they disagree in sign. Therefore the two-sided limit does not exist at all, not even as an infinite limit. Each one-sided infinite limit exists and conveys information, but they cannot be combined into a single two-sided limit when they go in opposite directions.
Question 5 Short Answer
Why does an infinite limit technically 'not exist' as a real number, and what useful information does writing lim(x→a) f(x) = ∞ still convey?
Think about your answer, then reveal below.
Model answer: The limit does not exist because ∞ is not a real number — the function never settles at a specific finite value. However, writing lim = ∞ communicates that the function grows without bound in the positive direction as x approaches a, which identifies a vertical asymptote and tells you the function's behavior is unbounded and in what direction. This is more informative than simply saying 'DNE.'
Distinguishing 'technically DNE' from 'DNE but in a specific infinite way' is crucial for rigorous analysis. The notation lim = ∞ is a shorthand for 'for every M > 0, there exists δ > 0 such that |x − a| < δ implies f(x) > M' — a precise statement about unbounded growth, not an assertion that the limit has value ∞. Keeping the technical DNE and the descriptive ∞ notation separate prevents the common error of treating infinity as an arithmetic constant.