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Satisfaction for ordered pairs

Source: Brazilian Math Olympiad - University Level - Galois-Noether 2018 #10

November 25, 2019
Brazilian Undergrad MOcollege contestlimitscalculus

Problem Statement

How many ordered pairs of real numbers (a,b) (a, b) satisfy equality lim⁡x→0sin⁡2xeax−2bx−1=12\lim_{x \to 0} \frac{\sin^2x}{e^{ax}-2bx-1}= \frac{1}{2}?
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