Home›Calculus 1›Limits at infinity›Problem 1.45

Limit of \( \displaystyle \frac{5 x - 2}{3 x + 7} \) as \( x \to \infty \)

Problem 1.45 · medium Video

Evaluate \( \displaystyle \lim_{x \to \infty} \frac{5 x - 2}{3 x + 7} \).

\[ \lim_{x \to \infty}\left(\frac{5 x - 2}{3 x + 7}\right) \]
limit algebraStart with the original limit expression. Divide both numerator and denominator by x.✓ Proved
\[ = \lim_{x \to \infty}\left(\frac{5 - \frac{2}{x}}{3 + \frac{7}{x}}\right) \]
simplifySimplify the fractions within the limit.✓ Proved
\[ = \lim_{x \to \infty} 5 \left(\lim_{x \to \infty} 3\right)^{-1} \]
limit-lawApply the limit laws for quotients and sums.✓ Proved
\[ = \frac{5}{3} \]
simplifyEvaluate the final numerical value.✓ Proved

Answer \( \frac{5}{3} \)

The graph of f(x) = (5x - 2)/(3x + 7). A point runs right along the curve; its height settles toward 5/3.

✓ 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 3*x + 7 = 0
3	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where 3*x + 7 = 0 undefined where 3 + 7/x = 0 undefined where x = 0
4	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where 3 + 7/x = 0 undefined where x = 0 undefined where Limit(3, x, oo, dir='-') = 0
5	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where Limit(3, 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: fail (error) — Step 4 incorrectly applies limit laws by splitting the limit of a quotient into a quotient of limits without first establishing that the denominator's limit is non-zero (it is 3, so it is valid, but the step also implicitly evaluates the inner limits 2/x and 7/x to 0 without showing that step, effectively combining limit evaluation and algebraic simplification). More critically, Step 4's notation `Limit(5 - 0, x, oo)` is syntactically weird and the transition from Step 3 to 4 skips the explicit evaluation of `Limit(2/x, x, oo)` and `Limit(7/x, x, oo)`. However, the primary defect is that Step 4 applies the quotient law `Limit(f/g) = Limit(f)/Limit(g)` which is valid, but the note says 'Apply the limit laws for quotients and sums'. The step actually performs the evaluation of the limits of the constant terms and the zero terms simultaneously. A stricter reading: Step 4 goes from `Limit((5 - 2/x)/(3 + 7/x))` to `Limit(5 - 0)/Limit(3 + 0)`. This step applies the quotient law AND substitutes the limits of `2/x` and `7/x` with 0. This is two rules at once (limit-law and substitution/infinity). Therefore, it violates the 'one rule per step' constraint.

Senior review claude-sonnet-5-5, 2026-10-03: fail — Every line is mathematically correct and SymPy verified each one, but step 4 bundles the limit laws with evaluating lim 2/x and lim 7/x = 0. It should be split into a limit-law step followed by an infinity step.

qwen3.6:27b-mlx: uphold — Step 4 applies the quotient and sum limit laws and also replaces 2/x and 7/x by 0 (a separate 'infinity' evaluation) in a single line, so it does two independent things; the math is still valid, since the denominator's limit is 3, not 0.

Every verdict on record (7)

qwen3.6:27b-mlx: fail (error) 2026-10-03 — Step 4 incorrectly applies limit laws by splitting the limit of a quotient into a quotient of limits without first establishing that the denominator's limit is non-zero (it is 3, so it is valid, but the step also implicitly evaluates the inner limits 2/x and 7/x to 0 without showing that step, effectively combining limit evaluation and algebraic simplification). More critically, Step 4's notation `Limit(5 - 0, x, oo)` is syntactically weird and the transition from Step 3 to 4 skips the explicit evaluation of `Limit(2/x, x, oo)` and `Limit(7/x, x, oo)`. However, the primary defect is that Step 4 applies the quotient law `Limit(f/g) = Limit(f)/Limit(g)` which is valid, but the note says 'Apply the limit laws for quotients and sums'. The step actually performs the evaluation of the limits of the constant terms and the zero terms simultaneously. A stricter reading: Step 4 goes from `Limit((5 - 2/x)/(3 + 7/x))` to `Limit(5 - 0)/Limit(3 + 0)`. This step applies the quotient law AND substitutes the limits of `2/x` and `7/x` with 0. This is two rules at once (limit-law and substitution/infinity). Therefore, it violates the 'one rule per step' constraint.
gpt-oss:20b: pass 2026-10-03
claude-sonnet-5-5: fail (style) 2026-10-03 — Every line is mathematically correct and SymPy verified each one, but step 4 bundles the limit laws with evaluating lim 2/x and lim 7/x = 0. It should be split into a limit-law step followed by an infinity step.
qwen3.6:27b-mlx: fail (error) 2026-09-29 — Step 4 incorrectly applies limit laws by splitting the limit of a quotient into a quotient of limits without verifying that the denominator's limit is non-zero (it is 3, so it is valid, but the step also implicitly evaluates the inner limits 2/x and 7/x to 0 simultaneously with the split, violating the 'one change per step' rule. Specifically, it combines the evaluation of the terms inside the limit with the application of the quotient law. A correct sequence would first evaluate the limits of the numerator and denominator separately or simplify the expression inside the limit to constants before applying the quotient law. However, the primary defect is that Step 4 performs multiple operations: it applies the quotient law AND evaluates the limits of 2/x and 7/x to 0 in the same step. The transition from Step 3 to Step 4 should either simplify the expression inside the limit to (5-0)/(3+0) using algebra/simplify, or apply limit laws to get Limit(5-2/x)/Limit(3+7/x). Step 4 does both: it splits the limit AND substitutes the values 0 for the vanishing terms. This is two rules at once.
gpt-oss:20b: pass 2026-09-29
qwen3.6:27b-mlx: fail (error) 2026-09-29 — Step 4 incorrectly applies limit laws by replacing the terms 2/x and 7/x with 0 inside the limit expression before evaluating the limit. The correct application of limit laws would separate the limits of the individual terms (e.g., Limit(5) - Limit(2/x)) and then evaluate those limits, rather than pre-evaluating the inner expressions to constants within the Limit operator.
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.