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**Hint**: The value of e is a constant number and also applied in physics. The value of e is named after the great mathematician named Leonhard Euler. Another name of the value of e is known as Napier’s constant. Thus, e is also a number as Euler’s constant too. Here, we will find the value of \[{e^\infty }\]and then using that value we will find the value of \[{e^{ - \infty }}\].

**:**

__Complete step-by-step answer__We will find the value of e when the power is infinity and then when the power is minus infinity.

Here, we will find the value of \[{e^\infty }\]as below:

Since, we know that e is a constant number (\[2.71828182845\])

\[ = {(2.71828182845)^\infty }\]

Since, a constant number multiplied by infinity times is equal to infinity

\[ = \infty \]

Thus, the value of \[{e^\infty } = \infty \].

Next, we will find the value of \[{e^{ - \infty }}\]as below:

\[{e^{ - \infty }}\]

\[ = \dfrac{1}{{{e^\infty }}}\]

We know that \[{e^\infty } = \infty \]and so substituting the value above, we will get,

\[ = \dfrac{1}{\infty }\]

\[ = 0\]

Thus, the value of \[{e^{ - \infty }} = 0\].

In short, when e is raised to power infinity, it means e is increasing at a very high rate and hence it is tending towards a very large number and hence we say that e raised to the power infinity tends to infinity. On the other hand when e is raised to the negative infinity then it becomes a very small number and hence tends to zero.

Hence, the value of \[{e^{ - \infty }} = 0\].

**So, the correct answer is “0”.**

**Note**: The value of exponential constant is known as e. The approximate value of e is 2.718, but the value of e is much larger than that. The complete value of e can go on for thousands of digits. The value of e is special and when it acts as the base of the logarithmic function, its value is 1. Below are some values of e when they are raised to certain power:

1) Value of e power one will be e \[ \Rightarrow {e^1} = e\]

2) Value of e power zero will be 1 \[ \Rightarrow {e^0} = 1\]

3) Value of e power infinity will be infinity \[ \Rightarrow {e^\infty } = \infty \]

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