Even though the problem is easy to solve with the use of L'Hôpital's rule, from an intuitive perspective I still find the result interesting.

While testing an approach to another problem, I started with the basic function:

$\displaystyle f_1(x)=e^x$

I wanted to modify it so that it would satisfy $\displaystyle f(0)=f'(0)=0$, and came up with:

$\displaystyle f_2(x)=e^x-x-1$

Then I wanted to modify it further so that it would intersect $\displaystyle (1,1)$ , and came up with:

$\displaystyle f_3(x)=\frac{e^x-x-1}{e-1-1}$

Finally, I wanted to add an argument to vary the curvature of the graph while maintaining the preceding requirements, so I changed it to:

$\displaystyle f_4(x)=\frac{e^{kx}-kx-1}{e^k-k-1}$

While playing around with different values of $\displaystyle k$ in a graphing app, I discovered (graphically, rather than analytically) that the function collimated with $\displaystyle x^2$ when $\displaystyle k\to0$ . That was completely unexpected, and, I thought, intriguing… regardless of how elementary the analytic proof is.