Limit of \( \displaystyle \frac{4 x^{2} - 10 x + 6}{2 x - 3} \) as \( x \to \frac{3}{2} \)
Problem 1.309 · medium
Evaluate \( \displaystyle \lim_{x \to \frac{3}{2}} \frac{4 x^{2} - 10 x + 6}{2 x - 3} \).
- \[ \lim_{x \to \frac{3}{2}^+}\left(\frac{4 x^{2} - 10 x + 6}{2 x - 3}\right) \]limitStart with the limit of the given function.✓ Proved
- \[ = \lim_{x \to \frac{3}{2}^+}\left(2 x - 2\right) \]factor cancelFactor the numerator. Cancel the common factor (2*x - 3).✓ Proved
- \[ = 1 \]substitution simplifySubstitute x = 3/2 into the simplified expression. Evaluate the final numerical value.✓ Proved
Answer \( 1 \)
✓ 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.
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 2*x - 3 = 0 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| 5 | ✓ 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 factors the numerator, cancels the common term, and evaluates the limit via substitution. Each step applies a single rule and is correctly labeled.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-06qwen3.6:27b-mlx: pass 2026-10-06 — The solution correctly factors the numerator, cancels the common term, and evaluates the limit via substitution. Each step applies a single rule and is correctly labeled.qwen3.6:27b-mlx: pass 2026-10-06gpt-oss:20b: pass 2026-10-06
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 2026-10-06 with SymPy 1.14.0.