I am trying to solve a high school exercise dealing with limits.
The limit I need to solve is $$l=\lim_{x\rightarrow 0}{\left(\frac{1+\sqrt{1+x^2}}{x}\right)^x}.$$
Which can be solved as follows: $$l=\lim_{x\rightarrow 0}{e^{\ln{\left(\frac{1+\sqrt{1+x^2}}{|x|}\right)^x}}}= \lim_{x\rightarrow 0}{e^{x\ln{\left(1+\sqrt{1+x^2}\right)}-x\ln{|x|}}}.$$
Since $\lim_{x\rightarrow 0}x\ln{x}=0$, then $l=1$.
Now, at this stage in the high school book the derivatives are not yet explained. Hence, no l’Hôpital theorem. Even reasonings about infinities are explained in the next chapter.
So, is there a way to solve the limit without using the $x\ln x$ limit, only by means of basic known limits?