Limit of \( \displaystyle \frac{\left(3 x + 1\right)^{2} - 9}{3 x + 4} \) as \( x \to - \frac{4}{3} \)
Problem 1.719 · medium Video
Evaluate \( \displaystyle \lim_{x \to - \frac{4}{3}} \frac{\left(3 x + 1\right)^{2} - 9}{3 x + 4} \).
- \[ \lim_{x \to - \frac{4}{3}^+}\left(\frac{\left(3 x + 1\right)^{2} - 9}{3 x + 4}\right) \]limitSet up the limit with the specified direction.✓ Proved
- \[ = \lim_{x \to - \frac{4}{3}^+}\left(\frac{9 x^{2} + 6 x - 8}{3 x + 4}\right) \]simplify simplifyExpand the numerator. Combine constant terms in the numerator.✓ Proved
- \[ = \lim_{x \to - \frac{4}{3}^+}\left(3 x - 2\right) \]factor cancelFactor the quadratic numerator. Cancel the common factor (3*x + 4).✓ Proved
- \[ = -6 \]limit limitEvaluate the remaining expression at x = -4/3. Calculate the final value.✓ Proved
Answer \( -6 \)
✓ Nihil obstat Every line of this solution was proved by the computer algebra system SymPy. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. How we verify
The full receipt
| Line | Status | Checked by | Detail |
|---|---|---|---|
| 1 | ✓ Proved | sympy 1.14.0 | line 1 is the problem as stated |
| 2 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 3*x + 4 = 0 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 3*x + 4 = 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 3*x + 4 = 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| 7 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| answer | ✓ Proved | sympy 1.14.0 | final line against the stated answer: simplify(a - b) reduced to 0 |
| answer, a second way | ✓ Proved | sympy 1.14.0 | SymPy took the limit from both sides and got the stated value |
Reviewers
gpt-oss:20b: passqwen3.6:27b-mlx: pass — The solution correctly applies algebraic simplification, factoring, and cancellation to resolve the indeterminate form. Each step changes only one aspect of the expression and uses valid labels from the fixed vocabulary.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-19 — The solution correctly applies algebraic simplification, factoring, and cancellation to resolve the indeterminate form. Each step changes only one aspect of the expression and uses valid labels from the fixed vocabulary.gpt-oss:20b: pass 2026-10-19qwen3.6:27b-mlx: pass 2026-10-18 — The solution correctly applies algebraic simplification, factoring, and cancellation to resolve the indeterminate form, followed by direct evaluation. Each step changes only one aspect of the expression and uses valid labels from the fixed vocabulary.gpt-oss:20b: pass 2026-10-18qwen3.6:27b-mlx: pass 2026-10-18 — The solution correctly applies algebraic simplification, factoring, and cancellation to resolve the indeterminate form, followed by direct evaluation. Each step changes only one aspect of the expression and uses valid labels from the vocabulary.gpt-oss:20b: pass 2026-10-18
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
gemma4:26b, checked 2026-10-19 with SymPy 1.14.0.