How to use exponential map to show that large spheres are close to flat?

Question

I want to show that if we have $S_r^n$ the n-sphere of radious $r$ in $\mathbb{R}^{n+1}$ and we let $r$ go to infinity, then the metric of $S_r^n$ converges to the metric of $\mathbb{R}^n$. I have to say that I'm not an expert in differential geometry. A friend of mine told me that using exponential map, but I don't know how.