Question

What is the value of $$e^{-\mathrm{\infty }}$$ ?

Answer 1

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 $$e^{\mathrm{\infty }}$$and then using that value we will find the value of $$e^{-\mathrm{\infty }}$$.

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 $$e^{\mathrm{\infty }}$$as below:
Since, we know that e is a constant number ($$2.71828182845$$)
$$=(2.71828182845{)}^{\mathrm{\infty }}$$
Since, a constant number multiplied by infinity times is equal to infinity
$$=\mathrm{\infty }$$
Thus, the value of $$e^{\mathrm{\infty }}=\mathrm{\infty }$$.
Next, we will find the value of $$e^{-\mathrm{\infty }}$$as below:
$$e^{-\mathrm{\infty }}$$
$$={\displaystyle \frac{1}{e^{\mathrm{\infty }}}}$$
We know that $$e^{\mathrm{\infty }}=\mathrm{\infty }$$and so substituting the value above, we will get,
$$={\displaystyle \frac{1}{\mathrm{\infty }}}$$
$$=0$$
Thus, the value of $$e^{-\mathrm{\infty }}=0$$.
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 $$e^{-\mathrm{\infty }}=0$$.
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 $$\Rightarrow e^{\mathrm{\infty }}=\mathrm{\infty }$$