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

Last updated date: 17th Jul 2024
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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 ${e^{ - \infty }}$.

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 $\Rightarrow {e^\infty } = \infty$