This limit represents the definition of the mathematical constant e, which is the base of natural logarithms.
It describes the limit:
limx β β (1 + 1/x)x
As x approaches infinity, this expression is known to converge to e, a fundamental constant in mathematics.
For \( \alpha, \beta, \gamma \in \mathbb{R} \), if \[ \lim_{x \to 0} \frac{x^2 \sin(\alpha x) + (\gamma - 1)e^{x^2}}{\sin(2x - \beta x)} = 3, \] then \( \beta + \gamma - \alpha \) is equal to:
If $\lim_{x \to 1} \frac{(x-1)(6+\lambda \cos(x-1)) + \mu \sin(1-x)}{(x-1)^3} = -1$, where $\lambda, \mu \in \mathbb{R}$, then $\lambda + \mu$ is equal to
The sum of the payoffs to the players in the Nash equilibrium of the following simultaneous game is ............
| Player Y | ||
|---|---|---|
| C | NC | |
| Player X | X: 50, Y: 50 | X: 40, Y: 30 |
| X: 30, Y: 40 | X: 20, Y: 20 | |