Let f(x)=lim n→∞ ∑ (tan(x/2^(r+1))+tan^3(x/2^(r+1))/1-tan^2(x/2^(r+1))) r ∈ to (r=0, n).

Question

Let \(f(\mathrm{x})=\lim\limits _{\mathrm{n} \rightarrow \infty} \sum\limits_{\mathrm{r}=0}^{\mathrm{n}}\left(\frac{\tan \left(\mathrm{x} / 2^{\mathrm{r}+1}\right)+\tan ^{3}\left(\mathrm{x} / 2^{\mathrm{r}+1}\right)}{1-\tan ^{2}\left(\mathrm{x} / 2^{\mathrm{r}+1}\right)}\right)\)

Then \(\lim\limits _{x \rightarrow 0} \frac{\mathrm{e}^{\mathrm{x}}-\mathrm{e}^{f(\mathrm{x})}}{(\mathrm{x}-f(\mathrm{x}))}\) is equal to _______.

Answer 1

Answer is: 1 

\(f(x)=\lim\limits _{n \rightarrow \infty} \sum\limits_{r=0}^{n}\left(\tan \frac{x}{2^{r}}-\tan \frac{x}{2^{r+1}}\right)=\tan x\)

\(\lim\limits _{x \rightarrow 0}\left(\frac{e^{x}-e^{\tan x}}{x-\tan x}\right)=\lim\limits _{x \rightarrow 0} e^{\tan x} \frac{\left(e^{x-\tan x}-1\right)}{(x-\tan x)}\)

= 1