It's well known that a sequence of two dimensional Riemannian manifolds with uniform sectional curvature lower bound can Gromov-Hausdorff converge to a cone. 

Let $y=|x|$, by rotating around the y-axis, we get a cone $X$. Now I want to construct functions $y=f_i(x)$ such that their rotations around the y-axis (denoted by $M_i$) are smooth Riemannian manifolds and Gromov-Hausdorff converge to the cone $X$. I also want to compute the Gaussian curvature of $M_i$.

I considered $f_i(x)=|x|^{a_i}$ such that $a_i>1$ decrease to 1, but $f_i''(0)=\infty$, so it's not smooth enough, and the curvature at $x=0$ can't be defined. 
I also considered mollification of $|x|$ by convolution of $\eta(x)=c\exp(\frac{1}{|x|^2-1})$, but we can't write down the explicit form of the integration. 

Can any one give good approximations such that $f_i(x)$ can explicit be written down?