Problem Analysis #19

GPT-5.1 (high) · Run 1

51.05s
6,113 toks
Problem Statement

For all x>0x > 0, let y1(x)y_1(x), y2(x)y_2(x), and y3(x)y_3(x) be the functions satisfying

dy1dx−(sin⁡x)2y1=0,y1(1)=5,\frac{dy_1}{dx} - (\sin x)^2 y_1 = 0, \quad y_1(1) = 5, dy2dx−(cos⁡x)2y2=0,y2(1)=13,\frac{dy_2}{dx} - (\cos x)^2 y_2 = 0, \quad y_2(1) = 13, dy3dx−(2−x3x3)y3=0,y3(1)=35e,\frac{dy_3}{dx} - \left(\frac{2 - x^3}{x^3}\right) y_3 = 0, \quad y_3(1) = 35e,

respectively. Find the value of

lim⁡x→0+y1(x)y2(x)y3(x)+989xx3esin⁡x.\lim_{x \to 0^+} \frac{y_1(x)y_2(x)y_3(x) + 989x}{x^3 e \sin x}.

\

Ground Truth
989
Execution Trace
Reasoning Process

[THOUGHT] I'll solve each differential equation symbolically, form the product y1y2y3, then analyze the limit numerically and symbolically using Python to confirm its behavior as x→0⁺. [/THOUGHT] [PYTHON] import sympy as sp

x = sp.symbols('x', positive=True)

Define the ODE solutions:

y1': y1' = (sin x)^2 * y1

integrand1 = sp.sin(x)**2 I1 = sp.integrate(integrand1, (x, 1, x)) # integral from 1 to x y1 = 5*sp.exp(I1)

y2': y2' = (cos x)^2 * y2

integrand2 = sp.cos(x)**2 I2 = sp.integrate(integrand2, (x, 1, x)) y2 = 13*sp.exp(I2)

y3': y3' = ((2 - x^3)/x^3) * y3

Integrate (2/x^3 - 1)

I3 = sp.integrate(2/x**3 - 1, x) C3 = sp.symbols('C3') y3_general = C3*sp.exp(I3)

Apply y3(1) = 35e

C3_value = sp.solve(sp.Eq(y3_general.subs(x, 1), 35*sp.E), C3)[0] y3 = sp.simplify(y3_general.subs(C3, C3_value))

print("y1(x) =", sp.simplify(y1)) print("y2(x) =", sp.simplify(y2)) print("y3(x) =", sp.simplify(y3))

Simplify product y1y2y3

prod_y = sp.simplify(y1y2y3) print("Product y1y2y3 =", prod_y)

Define the expression for the limit

expr = (prod_y + 989*x)/(x**3 * sp.E * sp.sin(x)) print("Expression =", sp.simplify(expr))

Study asymptotics numerically for small x

for val in [0.5, 0.2, 0.1, 0.05, 0.02, 0.01]: num_val = expr.subs(x, val).evalf() print(f"x = {val}, expr ≈ {num_val}")

Also, get a formal series of the product term near 0+

(using series expansion for the dominating 989x term)

series_sin = sp.series(sp.sin(x), x, 0, 5) print("sin(x) series:", series_sin)

Check behavior of the product term alone vs denominator

prod_term = prod_y/(x**3 * sp.E * sp.sin(x)) print("Product term at small x:") for val in [0.5, 0.2, 0.1, 0.05, 0.02, 0.01]: num_val = prod_term.subs(x, val).evalf() print(f"x = {val}, prod_term ≈ {num_val}")

Tool Output
[PYTHON OUTPUT] y1(x) = 5*exp(x/2 - sin(2*x)/4 - 1/2 + sin(2)/4) y2(x) = 13*exp(x/2 + sin(2*x)/4 - 1/2 - sin(2)/4) y3(x) = 35*exp(-x + 3 - 1/x**2) Product y1*y2*y3 = 2275*exp(2 - 1/x**2) Expression = (989*x + 2275*exp(2 - 1/x**2))*exp(-1)/(x**3*sin(x)) x = 0.5, expr ≈ 4925.59432783760 x = 0.2, expr ≈ 45783.7108390037 x = 0.1, expr ≈ 364439.863463409 x = 0.05, expr ≈ 2911875.26825908 x = 0.02, expr ≈ 45482127.9960437 x = 0.01, expr ≈ 363838831.268758 sin(x) series: x - x**3/6 + O(x**5) Product term at small x: x = 0.5, prod_term ≈ 1890.02164326333 x = 0.2, prod_term ≈ 0.0000540372226419187 x = 0.1, prod_term ≈ 2.30436758848711E-36 x = 0.05, prod_term ≈ 1.89576314216959E-165 x = 0.02, prod_term ≈ 7.09545247417316E-1076 x = 0.01, prod_term ≈ 7.02205276715279E-4332 [/PYTHON OUTPUT]
Reasoning Process

[THOUGHT] From the Python result, the expression simplifies to

y1(x)y2(x)y3(x)+989xx3esin⁡x=(989x+2275e2−1/x2)e−1x3sin⁡x.\frac{y_1(x)y_2(x)y_3(x) + 989x}{x^3 e \sin x} = \frac{(989x + 2275 e^{2 - 1/x^2})e^{-1}}{x^3 \sin x}.

As x→0+x \to 0^+, the term e2−1/x2→0e^{2 - 1/x^2} \to 0 super-exponentially, so the 2275e2−1/x22275 e^{2 - 1/x^2} part is negligible compared to 989x989x.

Thus, asymptotically,

(989x+2275e2−1/x2)e−1x3sin⁡x∼989x e−1x3sin⁡x.\frac{(989x + 2275 e^{2 - 1/x^2})e^{-1}}{x^3 \sin x} \sim \frac{989x\,e^{-1}}{x^3 \sin x}.

Since sin⁡x∼x\sin x \sim x as x→0x \to 0, the denominator behaves like x4x^4, so the whole expression behaves like

989x e−1x4=989e x3→+∞as x→0+.\frac{989x\,e^{-1}}{x^4} = \frac{989}{e\,x^3} \to +\infty \quad\text{as } x \to 0^+.

Therefore, the limit diverges to +∞+\infty. [/THOUGHT] ∞\boxed{\infty}

Final Model Output
\infty

Comparative Analysis (P19) (Average across runs)

Avg Time Taken

Gemini-3-Pro-Preview
894.8
K-EXAONE-236B-A23B
849.6
Solar-Open-100B
262.8
Deepseek-V3.2
239.3
K-EXAONE-236B-A23B
204.9
Llama-VARCO-8B-Instruct
162.1
Grok-4.1-fast
136.4
Solar-Pro-2 (31B)(high)
107.9
Kanana-2-30B-Thinking
76.7
Kanana-2-30B-Thinking-2601
68.5
GPT-5.1 (high)
51.0
A.X-4.0 (72B)
43.3
HCX-007(high)
37.2
GPT-5.2 (high)
32.8
Claude-Opus-4.5
29.2
EXAONE-4.0.1-32B (high)
5.3

Avg Token Usage

K-EXAONE-236B-A23B
70,061
K-EXAONE-236B-A23B
41,646
Solar-Open-100B
39,496
Gemini-3-Pro-Preview
28,664
Grok-4.1-fast
20,203
Solar-Pro-2 (31B)(high)
17,442
Kanana-2-30B-Thinking-2601
15,620
Deepseek-V3.2
14,643
Kanana-2-30B-Thinking
14,612
Llama-VARCO-8B-Instruct
11,147
A.X-4.0 (72B)
6,621
GPT-5.1 (high)
6,113
Claude-Opus-4.5
5,843
HCX-007(high)
4,628
GPT-5.2 (high)
4,356
EXAONE-4.0.1-32B (high)
4,114