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Measuring the limit of a point on a smooth.spline in R

I'm not sure if that's the right terminology.

I've entered some data into R, and I've put a smooth.spline through it using the following command.

smoothingSpline = smooth.spline(year, rate, spar=0.35)
plot(x,y)
lines(smoothingSpline)

Now I'd like to measure some limits (or where the curve is at a given x point), and maybe do some predictive analysis on points that extend beyond the graph.

Are there commands in R for measuring (or predicting) the points along a curve?