Limit of \( \displaystyle \frac{- 2 x + e^{2 x + 1} - 2}{\left(2 x + 1\right)^{2}} \) as \( x \to - \frac{1}{2} \)
Problem 1.822 · hard Video
Evaluate \( \displaystyle \lim_{x \to - \frac{1}{2}} \frac{- 2 x + e^{2 x + 1} - 2}{\left(2 x + 1\right)^{2}} \).
- \[ \lim_{x \to - \frac{1}{2}^+}\left(\frac{- 2 x + e^{2 x + 1} - 2}{\left(2 x + 1\right)^{2}}\right) \]limitEvaluate the limit of the given function.✓ Proved
- \[ = \lim_{x \to - \frac{1}{2}^+}\left(\frac{- 2 x + e^{2 x + 1} - 2}{4 \left(x + \frac{1}{2}\right)^{2}}\right) \]algebraRewrite the denominator by expanding (2x + 1)**2.✓ Proved
- \[ = \lim_{x \to - \frac{1}{2}^+}\left(\frac{\frac{d}{d x} \left(- 2 x + e^{2 x + 1} - 2\right)}{\frac{d}{d x} 4 \left(x + \frac{1}{2}\right)^{2}}\right) \]lhopitalApply L'Hopital's rule since the limit is of the form 0/0.✓ Proved
- \[ = \lim_{x \to - \frac{1}{2}^+}\left(\frac{2 e^{2 x + 1} - 2}{8 x + 4}\right) \]simplify substitutionCalculate the derivatives of the numerator and denominator. Let u = x + 1/2, so x = u - 1/2.✓ Proved
- \[ = \lim_{u \to 0^+}\left(\frac{2 e^{2 u} - 2}{8 u}\right) \]substitution factorRewrite the limit in terms of u. Factor out the constant 2 from the numerator.✓ Proved
- \[ = \lim_{u \to 0^+}\left(\frac{e^{2 u} - 1}{4 u}\right) \]simplifySimplify the fraction.✓ Proved
- \[ = \lim_{u \to 0^+}\left(\frac{\frac{d}{d u} \left(e^{2 u} - 1\right)}{\frac{d}{d u} 4 u}\right) \]lhopitalApply L'Hopital's rule again to the expression in terms of u.✓ Proved
- \[ = \lim_{u \to 0^+}\left(\frac{e^{2 u}}{2}\right) \]simplify simplifyCalculate the derivatives. Simplify the resulting fraction.✓ Proved
- \[ = \frac{1}{2} \]limitEvaluate the limit as u approaches 0.✓ Proved
Answer \( \frac{1}{2} \)
✓ 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 2*x + 1 = 0 undefined where x + 1/2 = 0 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where x + 1/2 = 0 undefined where Derivative(4*(x + 1/2)**2, x) = 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where Derivative(4*(x + 1/2)**2, x) = 0 undefined where 8*x + 4 = 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 8*x + 4 = 0 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 8*x + 4 = 0 undefined where u = 0 |
| 7 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where u = 0 |
| 8 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where u = 0 |
| 9 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where u = 0 undefined where Derivative(4*u, u) = 0 |
| 10 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where Derivative(4*u, u) = 0 |
| 11 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| 12 | ✓ 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 L'Hopital's rule and algebraic simplifications. 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 2026-11-09 — The solution correctly applies L'Hopital's rule and algebraic simplifications. Each step changes only one aspect of the expression and uses valid labels from the fixed vocabulary.gpt-oss:20b: pass 2026-11-09qwen3.6:27b-mlx: pass 2026-11-08 — The solution correctly applies L'Hopital's rule and algebraic simplifications. Each step changes only one aspect of the expression and uses valid labels from the fixed vocabulary.gpt-oss:20b: pass 2026-11-08qwen3.6:27b-mlx: pass 2026-11-08 — The solution correctly applies L'Hopital's rule and algebraic simplifications. Each step changes only one aspect of the expression and uses valid labels from the fixed vocabulary.gpt-oss:20b: pass 2026-11-08
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 2026-11-09 with SymPy 1.14.0.