How does $1/e^{(1/\infty)}$ equal $1$?

Question

My professor had us solve an improper integral, the problem or work I did correctly, however, when I got to the final part of the answer, I got $1/e^{1/\infty} - 1/e$.

My professor got this as well, but when she put the actual value, she put $1 - 1/e$.

I thought that $e^{(-1/\infty)}$ equals zero, because anything over infinity equals zero. Am I wrong? Why does it equal 1?

$\frac{1}{\infty}=0$ (assuming you have this definition) but $e^0=1$. — Isko10986, Nov 22 '16 at 02:50

score 5 · Answer 1 · answered Nov 22 '16 at 02:51

5

First of all, $\frac{1}{x^{1/\infty}}$ doesn't equal anything because $\infty$ is not a number.

However, $\lim_{x\to\infty} \frac{1}{e^{1/x}}=\frac{1}{e^0}=1$

answered Nov 22 '16 at 02:51

1

This is such an important answer. Students need to remember that $\infty$ is not a number and shouldn't just be substituted in. – Ian Miller Nov 22 '16 at 03:00

score 4 · Answer 2 · answered Nov 22 '16 at 02:50

4

HINT

Note that $(1)$ $\lim_{x\to \infty }1/x=0$, $(2)$ $e^0=1$, and $(3)$ the exponential function is continuous.

answered Nov 22 '16 at 02:50

Mark Viola

179,405

score -1 · Answer 3 · answered Nov 22 '16 at 02:50

-1

The bit that reads (-1/inf) is equivalent to zero. So now doing the exponential you have exp(0), which of course is 1.

answered Nov 22 '16 at 02:50

Penguino

1,169
6
10

How does $1/e^{(1/\infty)}$ equal $1$?

3 Answers3