Limit of \( \displaystyle \frac{e^{x - 1} - 1}{x - 1} \) as \( x \to 1 \)
Problem 1.231 · medium
Evaluate \( \displaystyle \lim_{x \to 1} \frac{e^{x - 1} - 1}{x - 1} \).
- \[ \lim_{x \to 1^+}\left(\frac{e^{x - 1} - 1}{x - 1}\right) \]limitStart with the limit of the function as x approaches 1 from the right.✓ Proved
- \[ = \lim_{x \to 1^+}\left(\frac{\frac{d}{d x} \left(e^{x - 1} - 1\right)}{\frac{d}{d x} \left(x - 1\right)}\right) \]lhopitalApply L'Hôpital's rule since the limit is in the indeterminate form 0/0.✓ Proved
- \[ = \lim_{x \to 1^+} e^{x - 1} \]simplifyDifferentiate the numerator and the denominator.✓ Proved
- \[ = 1 \]limit simplifyEvaluate the limit by substituting x = 1. Simplify the final 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 x - 1 = 0 undefined where Derivative(x - 1, x) = 0 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where Derivative(x - 1, x) = 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: fail (style) — Step 3 is labeled 'simplify' but performs differentiation, which is not a simplification rule. The differentiation should have been labeled 'derivative' in Step 2 or Step 3, but Step 2 already claimed 'lhopital' (which implies differentiation). The label 'simplify' in Step 3 is incorrect because the step computes derivatives, not just simplifies expressions.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (style) 2026-10-05 — Step 3 is labeled 'simplify' but performs differentiation, which is not a simplification rule. The differentiation should have been labeled 'derivative' in Step 2 or Step 3, but Step 2 already claimed 'lhopital' (which implies differentiation). The label 'simplify' in Step 3 is incorrect because the step computes derivatives, not just simplifies expressions.gpt-oss:20b: pass 2026-10-05qwen3.6:27b-mlx: pass 2026-10-05 — The solution correctly applies L'Hôpital's rule and simplifies the resulting expression. Each step adheres to the single-rule constraint and uses valid labels from the fixed vocabulary.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
gemma4:26b, checked 2026-10-05 with SymPy 1.14.0.