GRE question $\lim_{n \to \infty} \inf_{x \in \mathbb R} (e^x-nx)$

Question

How would you solve this question?

$$\lim_{n \to \infty} \inf_{x \in \mathbb R} (e^x-nx)$$

I found it in a GRE exam but I'm not sure how to solve it.

Where did you find that please? Include here if not yet there please — BCLC, Oct 26 '16 at 15:17

score 2 · Accepted Answer · answered Oct 25 '16 at 20:16

2

$\lim_{n\to\infty}\inf_{x\in\mathbb R}\left(e^x-nx\right) \leq \lim_{n\to\infty}\left(e^1-n\right)=-\infty$

answered Oct 25 '16 at 20:16

T.J. Gaffney

I don't understand why you chose x = 1 to get $(e^1 - n)$ , couldn't you just have easily chosen x = -1 to get $\lim_{n\rightarrow\infty} (e^{-1} + n) = \infty$? – lioumens Oct 25 '16 at 20:25
2

I could. But knowing that the limit is less than $\infty$ isn't very helpful. – T.J. Gaffney Oct 25 '16 at 20:27

1 Answers1