# Limit question $\lim\limits_{x \to \infty} \left(1-\frac{1}{x}\right)^\left(e^x\right)=1^\infty$indeterminate

In that problem

$$\lim\limits_{x \to \infty} \left(1-\frac{1}{x}\right)^\left(e^x\right)$$

I use $$\ln$$, then it gave me $$0\times\infty$$ indeterminate, then I use L'Hospital Rule but I cannot reach the any answer.

Sorry for grammatical mistakes, my native language is not English.

• Look at the logarithm. Commented Dec 7, 2019 at 11:08
– user
Commented Dec 7, 2019 at 11:15

HINT

Let

$$\left(1-\dfrac{1}{x}\right)^{(e^x)}= \left[\left(1-\dfrac{1}{x}\right)^{x}\right]^{\dfrac{e^x}x}$$

and use standard limits.

As an alternative we can use that

$$\left(1-\frac1x\right)^{e^x}=e^{e^x\log \left(1-\frac1x\right)}=e^{\frac{e^x}{x}\frac{\log \left(1-\frac1x\right)}{\frac1x}}$$

and use l’Hospital separately for the two parts

• $$\frac{e^x}{x}$$

• $$\frac{\log \left(1-\frac1x\right)}{\frac1x}$$

• After that,I used log but I could not find anything. Commented Dec 7, 2019 at 16:59
• @HuseyinOkanDemir We know that $\left(1-\dfrac{1}{x}\right)^{x}\to \frac 1e$, right?
– user
Commented Dec 7, 2019 at 17:01
• Sorry,I could not notice,thanks. Commented Dec 7, 2019 at 17:04
• @HuseyinOkanDemir Do not hesitate to ask for any clarification.
– user
Commented Dec 7, 2019 at 17:12
• How can I prove that $\left(1-\dfrac{1}{x}\right)^{x}\to \frac 1e$ Commented Dec 7, 2019 at 17:33

Hint:

Determine first the limit of the log with an asymptotic equivalent: $$\mathrm e^x\ln\Bigl(1-\frac 1x\Bigr)\sim_{x\to\infty} \mathrm e^x\cdot\Bigl(-\frac1x\Bigr).$$

Use $$0 \leq e^{e^x \ln(1-1/x)} \leq e^{-e^x /x}$$ and take the limit, which gives 0.

• I do not understand exactly, could you explain? Commented Dec 7, 2019 at 12:32
• @HuseyinOkanDemir the first expression is same as your function since $x=e^{lnx}$ for any positive $x$ Commented Dec 7, 2019 at 12:34
• I understood that,but I did not understand $\leq e^{-e^x /x}$ Commented Dec 7, 2019 at 13:04
• @HuseyinOkanDemir $\ln(1+x) \leq x$ for $x>-1$ . Commented Dec 7, 2019 at 13:07