What is the value of e to the power minus infinity?
The correct answer and explanation is:
The value of e−∞e^{-\infty} is 0.
Explanation:
The expression e−∞e^{-\infty} refers to the mathematical limit of the exponential function as the exponent approaches negative infinity. The number ee is the base of the natural logarithm, approximately equal to 2.71828. The exponential function exe^x describes how a quantity grows or decays depending on the value of xx.
When the exponent is negative, such as in e−∞e^{-\infty}, the value of the function decreases as xx becomes more negative. Specifically, as the exponent approaches negative infinity, the value of the function approaches 0.
To understand this more clearly, we can consider the following:
- When the exponent is zero, e0=1e^0 = 1.
- When the exponent is negative but finite, such as e−1e^{-1}, the value is 1/e1/e, which is a positive fraction.
- As the exponent becomes increasingly negative, the value of the exponential function continues to shrink, moving closer and closer to 0.
- Mathematically, we can express this as a limit: limx→−∞ex=0\lim_{x \to -\infty} e^x = 0
This means that as xx goes to negative infinity, exe^x approaches 0 but never actually reaches it. In practical terms, the function gets arbitrarily close to zero, but for any finite negative value of xx, exe^x is always a positive number, no matter how small.
Therefore, the value of e−∞e^{-\infty} is 0, indicating that as the exponent becomes more and more negative, the value of the function decays towards zero.