Limit of \( \displaystyle \frac{2 x - \sin{\left(2 x \right)}}{8 x^{3}} \) as \( x \to 0 \)
Problem 1.51 · medium Video
Evaluate \( \displaystyle \lim_{x \to 0} \frac{2 x - \sin{\left(2 x \right)}}{8 x^{3}} \).
- \[ \lim_{x \to 0^+}\left(\frac{2 x - \sin{\left(2 x \right)}}{8 x^{3}}\right) \]limitStart with the original limit.✓ Proved
- \[ = \lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \left(2 x - \sin{\left(2 x \right)}\right)}{\frac{d}{d x} 8 x^{3}}\right) \]lhopitalApply L'Hopital's rule because the limit is an indeterminate form 0/0.✓ Proved
- \[ = \lim_{x \to 0^+}\left(\frac{2 - 2 \cos{\left(2 x \right)}}{24 x^{2}}\right) \]simplify factorCompute the derivatives. Factor out the constant 2.✓ Proved
- \[ = \lim_{x \to 0^+}\left(\frac{1 - \cos{\left(2 x \right)}}{12 x^{2}}\right) \]simplifySimplify the fraction.✓ Proved
- \[ = \lim_{x \to 0^+}\left(\frac{2 x^{2} + O\left(x^{4}\right)}{12 x^{2}}\right) \]rewriteUse the Taylor series expansion for cos(2x).✓ Proved
- \[ = \lim_{x \to 0^+} \frac{1}{6} \]simplify algebra cancelKeep the leading term of the expansion. Expand the squared term. Cancel the x**2 terms.✓ Proved
- \[ = \frac{1}{6} \]simplifyFinal evaluation.✓ Proved
Answer \( \frac{1}{6} \)
✓ 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 x = 0 undefined where Derivative(8*x**3, x) = 0 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where Derivative(8*x**3, x) = 0 undefined where x = 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where x = 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where x = 0 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where x = 0 |
| 7 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where x = 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 |
| 10 | ✓ 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 Taylor series expansion to evaluate the limit. 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 2026-10-03 — The solution correctly applies L'Hopital's rule and Taylor series expansion to evaluate the limit. Each step changes only one aspect of the expression and uses valid labels from the fixed vocabulary.gpt-oss:20b: pass 2026-10-03qwen3.6:27b-mlx: pass 2026-09-29 — The solution correctly applies L'Hopital's rule and simplifies the resulting expression. The use of Taylor series expansion in step 6 is a valid method for evaluating the limit, and the subsequent simplification steps are algebraically sound.gpt-oss:20b: pass 2026-09-29qwen3.6:27b-mlx: pass 2026-09-29 — The solution correctly applies L'Hopital's rule and algebraic simplification. The use of Taylor series expansion in step 6 is a valid method for evaluating the resulting limit, and the subsequent steps correctly isolate the leading term.gpt-oss:20b: pass 2026-09-29
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 2026-10-03 with SymPy 1.14.0.