Home›Calculus 1›Limit laws with given limits›Problem 1.567

Limit laws with given limits

Problem 1.567 · easy

Suppose \( \displaystyle \lim_{x\to -1} f(x) = 4 \), \( \displaystyle \lim_{x\to -1} g(x) = 2 \) and \( \displaystyle \lim_{x\to -1} h(x) = 3 \). Find \( \displaystyle \lim_{x\to -1} \left(\sqrt{\left[f(x)\right]^{2} + \left[g(x)\right]^{2}}\right) \).

Each piece has a finite limit, and no denominator's limit is 0, so the limit laws (root) let us replace every function by its limit.
\[ \sqrt{2 \cdot 2 + 4 \cdot 4} = 2 \sqrt{5} \]
Substitute the three limits and simplify.✓ Proved

Answer \( 2 \sqrt{5} \)

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.

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 the limit of h(x) into the expression, which does not contain h(x). The problem asks for the limit involving only f(x) and g(x), so h(x) is irrelevant and its inclusion is a logical error.

Every verdict on record (4)

gpt-oss:20b: pass 2026-10-12
qwen3.6:27b-mlx: fail (error) 2026-10-12 — The solution incorrectly substitutes the limit of h(x) into the expression, which does not contain h(x). The problem asks for the limit involving only f(x) and g(x), so h(x) is irrelevant and its inclusion is a logical error.
gpt-oss:20b: pass 2026-10-12
qwen3.6:27b-mlx: pass 2026-10-12 — The solution correctly applies limit laws for sums, products, and roots. The substitution of limits into the expression is valid because the inner expression approaches a positive value, ensuring the square root is well-defined and continuous at that point.

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 2026-10-12 with SymPy 1.14.0.