Let $f(x,y) \rightarrow R$ be a function with continuous partial derivatives, and let $S$ be the surface $z=f(x,y)$ in $R^2$. Let $P_{0}=(x_0,y_0,z_0 )$ be a point in S and $P=(x,y,z)$ be some other point in S. We're asked to show that $a$, the angle between the plane tangent to S at $(x_0,y_0,z_0)$ and the vector $P-P_0$, approaches $0$ as $P\rightarrow P_0$.
I'd appreciate some help with proving this. The question is from a former exam in my multivariable calculus course.
The angle thing ticks me off here since I have no idea how to approach this. I tried proving this using dot products to derive the angle between the vectors, but calculating the limit was difficult.
Thanks!