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Limit laws with given limits

Problem 1.780 · easy

Suppose \( \displaystyle \lim_{x\to -4} f(x) = 8 \), \( \displaystyle \lim_{x\to -4} g(x) = -3 \) and \( \displaystyle \lim_{x\to -4} h(x) = -2 \). Find \( \displaystyle \lim_{x\to -4} \left(\left(f(x) - g(x)\right)^{2}\right) \).

Each piece has a finite limit, and no denominator's limit is 0, so the limit laws (power) let us replace every function by its limit.
\[ \left(3 + 8\right)^{2} = 121 \]
Substitute the three limits and simplify.✓ Proved

Answer \( 121 \)

Lines: 1 proved, 1 not checked. The answer was also checked a second way, without looking at the solution. The reviewers disagree about how one step is explained; every verdict is in the receipt. How we verify

The full receipt

Line	Status	Checked by	Detail
1	Not checked	—	a sentence; read, not computed
2	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
answer, a second way	✓ Checked independently	sympy 1.14.0 + mpmath 1.3.0	the limit of the same combination of concrete functions with those limits, taken by SymPy

Reviewers

gpt-oss:20b: pass
qwen3.6:27b-mlx: fail (error) — The solution incorrectly substitutes g(x) as 3 instead of -3, leading to (8 - 3)^2 = 25 instead of the correct (8 - (-3))^2 = 121. Although the final answer is coincidentally correct, the intermediate calculation shown is wrong.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-10-31 — The solution incorrectly substitutes g(x) as 3 instead of -3, leading to (8 - 3)^2 = 25 instead of the correct (8 - (-3))^2 = 121. Although the final answer is coincidentally correct, the intermediate calculation shown is wrong.
gpt-oss:20b: pass 2026-10-31
qwen3.6:27b-mlx: fail (error) 2026-10-30 — The substitution in step 2 is incorrect; it calculates (8 - (-3))^2 = 11^2 = 121, but writes the expression as (3 + 8)**2, implying g(x) was substituted as 3 instead of -3, or the subtraction was handled incorrectly in the text representation. While the final number 121 is correct for (8 - (-3))^2, the intermediate expression (3+8) suggests a sign error in the setup (treating -g as +3) which is misleading. Wait, 8 - (-3) is 11. 3+8 is 11. The text says 'Substitute the three limits'. If it substitutes g=-3, it should be (8 - (-3))^2. Writing (3+8) implies it substituted g=3 or converted -g to +3 without explanation. This is a 'misleading' or 'error' in the explanation of the step. Let's look closer. The problem asks for (f-g)^2. f->8, g->-3. f-g -> 8 - (-3) = 11. 11^2 = 121. The solution writes (3+8)**2. This is mathematically equivalent to 11^2, but it obscures the fact that g is -3. It looks like the solver treated the term as f + |g| or simply made a sign error in writing the step but got the right magnitude. However, strictly speaking, substituting g=-3 into f-g yields 8 - (-3). Writing 3+8 is not the direct substitution. It's a 'style' or 'misleading' issue if the student thinks g=3. But actually, looking at the previous line 'Each piece has a finite limit...', the sentence is fine. The equation line is marked 'proved' by CAS. The CAS checked (3+8)**2 = 121. The issue is whether the sentence supports the equation. The sentence says 'replace every function by its limit'. Replacing f by 8 and g by -3 gives (8 - (-3))^2. The equation shows (3+8)**2. This is a discrepancy between the stated method (substitution) and the executed step (which looks like a sign error or simplification not shown). A student might learn that -(-3) becomes +3, which is correct, but the notation (3+8) is ambiguous. Is it an error? If I write 8 - (-3) = 8 + 3 = 11, then 11^2. The solution skips the 8+3 step and goes to 3+8. This is acceptable arithmetic. However, usually, one writes (8 - (-3))^2. Writing (3+8) suggests the solver might have thought g=3. Given the adversarial nature, this is 'misleading' because it hides the sign of g. But is it an 'error'? The result is correct. The step is arithmetically valid if interpreted as 8+3. But the prompt says 'Substitute the three limits'. Substituting g=-3 into f-g gives 8-(-3). The equation shows 3+8. This is not the direct substitution. It's a simplified form. I will mark it as 'misleading' because it obscures the sign of the limit of g, potentially teaching a student to ignore signs or that -(-3) is just 3 without showing the operation. Actually, let's look at the options. 'error' is for wrong steps. 'misleading' is for false lessons. 'style' is for clarity. This is borderline. Let's look at the sentence again. 'replace every function by its limit'. If I replace g by -3, I get 8 - (-3). The equation has 3. This is a mismatch. The equation does not reflect the substitution described. It reflects a simplification. If the solution claimed 'Substitute... and simplify', it's okay. But it just says 'Substitute... and simplify'. The jump from 8-(-3) to 3+8 is a simplification. It's not wrong. But it's weird. Let's check if there is a more serious error. No. The answer 121 is correct. The method is correct. The representation is slightly sloppy. I will call it 'style' or 'misleading'. Let's go with 'misleading' because a student might think g(x) approaches 3.
gpt-oss:20b: pass 2026-10-30
qwen3.6:27b-mlx: fail (error) 2026-10-30 — The solution incorrectly substitutes g(x) as 3 instead of -3. Since lim g(x) = -3, the term (f(x) - g(x)) becomes (8 - (-3)) = 11, not (8 - 3) = 5. The correct limit is 121, but the arithmetic shown in step 2 (3+8)^2 implies an error in sign handling or transcription, even if the final number happens to be correct due to a coincidence in the problem statement vs the solution's logic (wait, 11^2 is 121, 5^2 is 25. The solution claims (3+8)^2 = 121, which is arithmetically false: 11^2=121, but 3+8=11. Ah, the solution wrote (3+8)^2. If they meant f-g = 8 - (-3) = 11, then 11^2=121. But they wrote 3+8. This suggests they treated g as 3 and added it? Or they treated -g as 3? Let's look closer. f-g = 8 - (-3) = 11. The solution writes (3+8)^2. This equals 121. So the arithmetic 11^2=121 is correct. But where did the 3 come from? g is -3. So -g is 3. So f - g = f + (-g) = 8 + 3 = 11. The solution writes (3+8)^2. This is technically correct if interpreted as (-g + f). However, the text says 'Substitute the three limits'. It does not explicitly show the sign change for g. It just says 'Substitute... and simplify'. The equation line is `(3 + 8)**2 = 121`. This is ambiguous. Did they substitute g=-3 and do 8 - (-3)? Or did they substitute g=3? If they substituted g=3, they would get 8-3=5, 5^2=25. They got 121. So they must have done 8+3. But the expression is f-g. So they effectively computed f+|g| or f+(-g). The sentence says 'replace every function by its limit'. It does not explain the sign flip. A student might think lim(f-g) = lim f - lim g = 8 - 3 = 5. The solution's equation line is `(3+8)**2`. This is confusing. It looks like they substituted 3 for g? No, 3+8=11. If they substituted 3 for g, they would have 8-3. If they substituted -3 for g, they would have 8-(-3)=11. The equation shows 3+8. This implies they replaced -g with 3. This is a valid algebraic step, but the sentence 'replace every function by its limit' is slightly misleading if it doesn't account for the operator. However, the bigger issue is that the equation line is marked 'unchecked' in the prompt description but the prompt says 'Each line... is either an EQUATION... checked... or a SENTENCE'. The prompt says '2. [equation, unchecked]'. This is a contradiction in the prompt's meta-data or my reading. Let's assume the equation is provided as is. The equation `(3 + 8)**2 = 121` is arithmetically true. But does it follow from the limits? lim f = 8, lim g = -3. lim (f-g)^2 = (8 - (-3))^2 = 11^2 = 121. The solution writes (3+8)^2. This is equivalent to (8+3)^2. This is correct. Is there an error? The sentence says 'no denominator's limit is 0'. This is irrelevant for a polynomial expression, but not wrong. The main risk is clarity. Did the student make a sign error and get lucky? 8-3=5, 5^2=25. 8-(-3)=11, 11^2=121. The answer 121 is correct. The intermediate step (3+8) suggests they converted -g to +3. This is correct. I will mark it as pass because the math is correct, even if the notation '3' for '-g' is slightly opaque without context. Wait, looking at the provided solution again. 'Substitute the three limits'. If I substitute g=-3 into f-g, I get 8-(-3). The solution writes 3+8. This is a simplification step. It is correct. I will pass it.
gpt-oss:20b: pass 2026-10-30

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 generator:structured/limit_laws_values, checked 2026-10-31 with SymPy 1.14.0.