Home›Calculus 1›Limits at infinity›Problem 1.27

Limit of \( \displaystyle \frac{x^{3} - x}{2 x^{3} + 5} \) as \( x \to \infty \)

Problem 1.27 · medium Video

Evaluate \( \displaystyle \lim_{x \to \infty} \frac{x^{3} - x}{2 x^{3} + 5} \).

\[ \lim_{x \to \infty}\left(\frac{x^{3} - x}{2 x^{3} + 5}\right) \]
limitStart with the original limit expression.✓ Proved
\[ = \lim_{x \to \infty}\left(\frac{1 - \frac{1}{x^{2}}}{2 + \frac{5}{x^{3}}}\right) \]
factor cancelFactor out the highest power of x from the numerator and denominator. Cancel the common factor x**3.✓ Proved
\[ = \lim_{x \to \infty}\left(1 - \frac{1}{x^{2}}\right) \left(\lim_{x \to \infty}\left(2 + \frac{5}{x^{3}}\right)\right)^{-1} \]
limit-lawApply the quotient rule for limits.✓ Proved
\[ = \lim_{x \to \infty} 1 \left(\lim_{x \to \infty} 2\right)^{-1} \]
limit-lawEvaluate the limits of the individual terms.✓ Proved
\[ = \frac{1}{2} \]
limitSimplify the final result.✓ Proved

Answer \( \frac{1}{2} \)

The graph of f(x) = (x³ - x)/(2x³ + 5). A point runs right along the curve; its height settles toward 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 2x3 + 5 = 0 undefined where 2 + 5/x*3 = 0 undefined where x = 0
3	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where 2 + 5/x**3 = 0 undefined where x = 0
4	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where 2 + 5/x3 = 0 undefined where x = 0 undefined where Limit(2 + 5/x3, x, oo, dir='-') = 0
5	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where Limit(2 + 5/x**3, x, oo, dir='-') = 0 undefined where x = 0 undefined where Limit(2, x, oo, dir='-') = 0
6	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where Limit(2, x, oo, dir='-') = 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: pass
qwen3.6:27b-mlx: pass

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-03
gpt-oss:20b: pass 2026-10-03
qwen3.6:27b-mlx: pass 2026-09-29
gpt-oss:20b: pass 2026-09-29
qwen3.6:27b-mlx: pass 2026-09-29 — The solution correctly applies factoring, cancellation, and limit laws 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 2026-09-29

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.