In a previous problem, I showed that $f_n\left(x\right)=n\left(\sqrt[n]{x}-1\right)$ converges pointwise to $\ln x$ on $\left(0,\infty\right)$ and uniformly on $\left[e^{-A},e^A\right]$. I am now trying to show that its inverse, $f_n^{-1}\left(x\right)=\left(x/n+1\right)^n$, converges pointwise to $f^{-1}\left(x\right)=e^x$ on $\mathbb{R}$ and uniformly on $\left[-A,A\right]$.
Now, what I have in mind is the following: $\lim_{n\to\infty}\left(\frac{x}{n}+1\right)^n=\exp\left(\lim_{n\to\infty}n\ln\left(\frac{x}{n}+1\right)\right).$
If we let $t=1/n$, then we get $\exp\left(\lim_{t\to0}\frac{1}{t}\ln\left(tx+1\right)\right)=\exp\left(\frac{0}{0}\right).$
Therefore, we can apply L'Hôpital's rule: $\exp\left(\lim_{t\to0}\frac{x}{tx+1}\right)=e^x.$
Is this correct? I have a feeling that, in the first step, I may not exponentiate a limit like that.
Furthermore, when it comes to showing uniform convergence, I know that I need to prove that for every $\epsilon>0$, there is an $N\in\mathbb{N}$ such that $\left|f_n^{-1}(x)-f^{-1}(x)\right|=\left|\left(\frac{x}{n}+1\right)^n-e^x\right|<\epsilon$whenever $n\geq N$ and $x\in[-A,A]$.
However, in that last problem, I had to use an esoteric definition of the exponential, and I am afraid that I may need to do the same for this problem. Do you guys have any ideas? Thanks a whole lot!
By the way, the book hints that I use the MVT, but I am clueless as to how to apply it here.