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

Question

Vedantu Content Team · Accepted Answer

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