One-sided limits of piecewise functions
Problem 1.268 · easy
Let \( \displaystyle f(x) = \begin{cases} - 3 x^{2} + x + 2 & x < 3 \\ 0 & x = 3 \\ - x^{2} - x + 3 & x > 3 \end{cases} \). Find \( \displaystyle \lim_{x\to 3^-} f(x) \), \( \displaystyle \lim_{x\to 3^+} f(x) \), \( \displaystyle \lim_{x\to 3} f(x) \) and \( \displaystyle f(3) \).
- For x < 3 only the first piece matters, so the left-hand limit is the limit of that polynomial.Reviewed
- \[ \lim_{x \to 3^-}\left(- 3 x^{2} + x + 2\right) = -22 \]Left-hand limit.✓ Proved
- \[ \lim_{x \to 3^+}\left(- x^{2} - x + 3\right) = -9 \]Right-hand limit.✓ Proved
- The one-sided limits are different, so the two-sided limit does not exist.Reviewed
- The value f(3) = 0 is read straight from the middle line; it does not affect any of the limits.Reviewed
Answer \( \lim_{x\to 3^-} f(x) = -22,\ \lim_{x\to 3^+} f(x) = -9,\ \lim_{x\to a} f(x) \text{ does not exist},\ f(3) = 0 \)
✓ Nihil obstat Lines: 2 proved, 3 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation.
The full receipt
| Line | Status | Checked by | Detail |
|---|---|---|---|
| 1 | Reviewed | reviewers: gpt-oss:20b, qwen3.6:27b-mlx | a sentence; read, not computed |
| 2 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 4 | Reviewed | reviewers: gpt-oss:20b, qwen3.6:27b-mlx | a sentence; read, not computed |
| 5 | Reviewed | reviewers: gpt-oss:20b, qwen3.6:27b-mlx | a sentence; read, not computed |
| answer, a second way | ✓ Checked independently | sympy 1.14.0 + mpmath 1.3.0 | the pieces evaluated at a ± 1e-12 |
Reviewers
gpt-oss:20b: passqwen3.6:27b-mlx: pass
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-05gpt-oss:20b: pass 2026-10-05qwen3.6:27b-mlx: pass 2026-10-05 — The solution correctly identifies the relevant pieces for the one-sided limits, computes them accurately, and correctly concludes that the two-sided limit does not exist because the one-sided limits differ. The value of f(3) is also correctly identified.gpt-oss:20b: pass 2026-10-05
Proved: SymPy reduced the difference between the two sides to zero. Checked independently: a separate
method, named above, confirmed it. Checked numerically: the two sides agree at every sampled point, which is evidence,
not proof. Reviewed: a model or a person read it; that is all a sentence can have. Solution by
generator:structured/one_sided_limits, checked 2026-10-05 with SymPy 1.14.0.