Home›Calculus 1›Limits at infinity›Problem 1.365

Limit of \( \displaystyle \frac{5 x + 3}{3 x + 10} \) as \( x \to \infty \)

Problem 1.365 · medium

Evaluate \( \displaystyle \lim_{x \to \infty} \frac{5 x + 3}{3 x + 10} \).

\[ \lim_{x \to \infty}\left(\frac{5 x + 3}{3 x + 10}\right) \]
limitStart with the original limit expression.✓ Proved
\[ = \lim_{x \to \infty}\left(\frac{5 + \frac{3}{x}}{3 + \frac{10}{x}}\right) \]
algebra simplifyDivide both numerator and denominator by x. Simplify the fractions inside the limit.✓ Proved
\[ = \lim_{x \to \infty}\left(5 + \frac{3}{x}\right) \left(\lim_{x \to \infty}\left(3 + \frac{10}{x}\right)\right)^{-1} \]
limit-lawApply the quotient rule for limits.✓ Proved
\[ = \lim_{x \to \infty} 5 \left(\lim_{x \to \infty} 3\right)^{-1} \]
limit-lawEvaluate the limits of the individual terms.✓ Proved
\[ = \frac{5}{3} \]
limitThe final value of the limit.✓ Proved

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

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 undefined where 3*x + 10 = 0 undefined where x = 0 undefined where 3 + 10/x = 0
3	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where x = 0 undefined where 3 + 10/x = 0
4	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where x = 0 undefined where 3 + 10/x = 0 undefined where Limit(3 + 10/x, x, oo, dir='-') = 0
5	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where x = 0 undefined where Limit(3 + 10/x, x, oo, dir='-') = 0 undefined where Limit(3, x, oo, dir='-') = 0
6	✓ 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 5 is mathematically incorrect. It claims that Limit(5 + 3/x, x, oo) simplifies to Limit(5, x, oo) via a 'limit-law', but it ignores the term 3/x. The correct application of limit laws would require evaluating Limit(3/x, x, oo) = 0 separately and adding it to 5, or simplifying the expression inside the limit first. As written, it asserts that the limit of a sum is the limit of one part, which is false.

Senior review claude-sonnet-5-5, 2026-10-07: fail — The final answer 5/3 is correct and every line is equal in value. Line 5 silently drops the vanishing terms 3/x and 10/x, with a note that says 'evaluate the limits of the individual terms'. A student could read this as dropping terms inside a limit. A correct solution would split the sums and then evaluate Limit(3/x)=0 and Limit(10/x)=0 as separate steps.

qwen3.6:27b-mlx: uphold — The line is true in value (SymPy proved it), but going from Limit(5+3/x)/Limit(3+10/x) to Limit(5)/Limit(3) compresses the sum law and the evaluations 3/x→0 and 10/x→0 into one step labelled limit-law, so it genuinely does more than one thing.

Every verdict on record (5)

gpt-oss:20b: pass 2026-10-07
qwen3.6:27b-mlx: fail (error) 2026-10-07 — Step 5 is mathematically incorrect. It claims that Limit(5 + 3/x, x, oo) simplifies to Limit(5, x, oo) via a 'limit-law', but it ignores the term 3/x. The correct application of limit laws would require evaluating Limit(3/x, x, oo) = 0 separately and adding it to 5, or simplifying the expression inside the limit first. As written, it asserts that the limit of a sum is the limit of one part, which is false.
claude-sonnet-5-5: fail (misleading) 2026-10-07 — The final answer 5/3 is correct and every line is equal in value. Line 5 silently drops the vanishing terms 3/x and 10/x, with a note that says 'evaluate the limits of the individual terms'. A student could read this as dropping terms inside a limit. A correct solution would split the sums and then evaluate Limit(3/x)=0 and Limit(10/x)=0 as separate steps.
gpt-oss:20b: pass 2026-10-07
qwen3.6:27b-mlx: fail (error) 2026-10-07 — Step 5 is mathematically incorrect because it claims Limit(5 + 3/x) equals Limit(5), ignoring the term 3/x. While the final answer is correct, the step skips the evaluation of the vanishing terms (3/x and 10/x) without justification, effectively asserting that Limit(3/x) is zero without showing it or applying a limit law to separate the constant from the variable term.

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.