Home›Calculus 1›One-sided limits of piecewise functions›Problem 1.516

One-sided limits of piecewise functions

Problem 1.516 · easy

Let \( \displaystyle f(x) = \begin{cases} x - 4 & x < -2 \\ 0 & x = -2 \\ - 3 x^{2} - 3 x & x > -2 \end{cases} \). Find \( \displaystyle \lim_{x\to -2^-} f(x) \), \( \displaystyle \lim_{x\to -2^+} f(x) \), \( \displaystyle \lim_{x\to -2} f(x) \) and \( \displaystyle f(-2) \).

For x < -2 only the first piece matters, so the left-hand limit is the limit of that polynomial.
Reviewed
\[ \lim_{x \to -2^-}\left(x - 4\right) = -6 \]
Left-hand limit.✓ Proved
\[ \lim_{x \to -2^+}\left(- 3 x^{2} - 3 x\right) = -6 \]
Right-hand limit.✓ Proved
The two one-sided limits agree, so the limit exists and equals their common value.
Reviewed
The value f(-2) = 0 is read straight from the middle line; it does not affect any of the limits.
Reviewed

Answer \( \lim_{x\to -2^-} f(x) = -6,\ \lim_{x\to -2^+} f(x) = -6,\ \lim_{x\to -2} f(x) = -6,\ f(-2) = 0 \)

✓ 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: pass
qwen3.6:27b-mlx: pass — The solution correctly identifies the relevant pieces for the one-sided limits and the function value. The logic regarding the existence of the limit and the independence of the function value from the limit is sound.

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 and the function value. The logic regarding the existence of the limit and the independence of the function value from the limit is sound.
qwen3.6:27b-mlx: pass 2026-10-10 — The solution correctly identifies the relevant pieces for the one-sided limits and the function value. The reasoning is sound and the calculations are correct.
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.