Home›Calculus 1›One-sided limits of piecewise functions›Problem 1.520

One-sided limits of piecewise functions

Problem 1.520 · easy

Let \( \displaystyle f(x) = \begin{cases} - x^{2} - 2 x & x < 3 \\ -1 & x = 3 \\ - 2 x^{2} + 3 x + 2 & x > 3 \end{cases} \). 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
\[ \lim_{x \to 3^-}\left(- x^{2} - 2 x\right) = -15 \]
Left-hand limit.✓ Proved
\[ \lim_{x \to 3^+}\left(- 2 x^{2} + 3 x + 2\right) = -7 \]
Right-hand limit.✓ Proved
The one-sided limits are different, so the two-sided limit does not exist.
Reviewed
The value f(3) = -1 is read straight from the middle line; it does not affect any of the limits.
Reviewed

Answer \( \lim_{x\to 3^-} f(x) = -15,\ \lim_{x\to 3^+} f(x) = -7,\ \lim_{x\to a} f(x) \text{ does not exist},\ f(3) = -1 \)

✓ 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: pass
qwen3.6:27b-mlx: pass — The solution correctly identifies the relevant pieces for the one-sided limits, verifies the calculations, and correctly concludes that the two-sided limit does not exist due to the mismatch. The distinction between the function value and the limit is also handled correctly.

Every verdict on record (4)

gpt-oss:20b: pass 2026-10-10
qwen3.6:27b-mlx: pass 2026-10-10 — The solution correctly identifies the relevant pieces for the one-sided limits, verifies the calculations, and correctly concludes that the two-sided limit does not exist due to the mismatch. The distinction between the function value and the limit is also handled correctly.
qwen3.6:27b-mlx: pass 2026-10-10 — 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 correctly identified as independent of the limits.
gpt-oss:20b: pass 2026-10-10

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 2026-10-10 with SymPy 1.14.0.