Limit of \( \displaystyle \left(3 x - 1\right) e^{1 - 3 x} \) as \( x \to \infty \)
Problem 1.361 · medium
Evaluate \( \displaystyle \lim_{x \to \infty} \left(3 x - 1\right) e^{1 - 3 x} \).
- \[ \lim_{x \to \infty}\left(\left(3 x - 1\right) e^{1 - 3 x}\right) \]limitEvaluate the limit as x approaches infinity.✓ Proved
- \[ = \lim_{x \to \infty}\left(e \left(3 x - 1\right) e^{- 3 x}\right) \]rewrite rewriteSeparate the exponential term. Rewrite the negative exponent as a fraction.✓ Proved
- \[ = \lim_{x \to \infty}\left(e \lim_{x \to \infty}\left(\frac{\frac{d}{d x} \left(3 x - 1\right)}{\frac{d}{d x} e^{3 x}}\right)\right) \]lhopitalApply L'Hopital's Rule to the indeterminate form.✓ Proved
- \[ = \lim_{x \to \infty}\left(e e^{- 3 x}\right) \]simplify simplify rewriteCompute the derivatives. Simplify the fraction. Rewrite the expression using a negative exponent.✓ Proved
- \[ = \lim_{x \to \infty} 0 \]limitEvaluate the limit of the exponential term.✓ Proved
- \[ = 0 \]limitFinal result.✓ Proved
Answer \( 0 \)
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. A reviewer's objection to the explanation stands; see the receipt.
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 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where Derivative(exp(3*x), x) = 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where Derivative(exp(3*x), x) = 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 |
| 8 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| 9 | ✓ 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: fail (error) — Step 4 applies L’Hôpital’s rule and pulls out the constant exp(1) in a single transition, changing two aspects at once. The step should either apply only the rule or separate the factoring of exp(1) into its own step.qwen3.6:27b-mlx: fail (error) — Step 4 incorrectly applies L'Hopital's Rule by introducing a nested limit structure `Limit(exp(1)*Limit(...))` which is not the standard application of the rule and implies the limit of the ratio of derivatives is a separate entity being evaluated inside the outer limit, rather than replacing the original limit directly. Furthermore, the notation `Limit(Derivative(...)/Derivative(...), x, oo)` suggests the derivatives are functions of x inside a limit, but the subsequent step 5 simplifies them to constants/functions without showing the limit evaluation process clearly, making the transition from step 4 to 5 logically disjointed and formally incorrect.
Every verdict on record (4)
gpt-oss:20b: fail (error) 2026-10-07 — Step 4 applies L’Hôpital’s rule and pulls out the constant exp(1) in a single transition, changing two aspects at once. The step should either apply only the rule or separate the factoring of exp(1) into its own step.qwen3.6:27b-mlx: fail (error) 2026-10-07 — Step 4 incorrectly applies L'Hopital's Rule by introducing a nested limit structure `Limit(exp(1)*Limit(...))` which is not the standard application of the rule and implies the limit of the ratio of derivatives is a separate entity being evaluated inside the outer limit, rather than replacing the original limit directly. Furthermore, the notation `Limit(Derivative(...)/Derivative(...), x, oo)` suggests the derivatives are functions of x inside a limit, but the subsequent step 5 simplifies them to constants/functions without showing the limit evaluation process clearly, making the transition from step 4 to 5 logically disjointed and formally incorrect.gpt-oss:20b: fail (error) 2026-10-07 — Step 4 applies two rules at once: it rewrites the product of a constant and a limit as a nested limit and then applies L’Hôpital’s rule. The correct approach would keep the constant outside the limit and apply L’Hôpital to the whole fraction, not introduce an extra nested limit. This violates the one‑rule‑per‑step rule and misrepresents the application of L’Hôpital’s rule.qwen3.6:27b-mlx: fail (error) 2026-10-07 — Step 4 incorrectly applies L'Hopital's Rule by nesting the limit operator inside the expression (Limit(exp(1)*Limit(...))), which is syntactically and logically invalid; L'Hopital's Rule applies to the ratio inside the limit, not by introducing a new limit operator within the expression.
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 2026-10-07 with SymPy 1.14.0.