3.1. CONTINUITY AND LIMITS 221 2. For every ε >0 there exists δ >0 such that for points −→ x ∈ D0 < dist( −→ x , −→ x0 ) < δ guarantees | f ( −→ x ) − L | < ε. The ε-δ formulation can sometimes be awkward to apply, but for Fnding limits of functions of two variables at the origin in R 2 , we can sometimes use a related trick, based on polar coordinates. To see how it works, consider the example f ( x,y ) = x 3 x 2 + y 2 , ( x,y ) n = (0 , 0) . If we express this in the polar coordinates of ( x,y ) x = r cos θ y = r sin θ we have f ( r cos θ,r sin θ ) = r 3 cos 3 θ r 2 cos 2 + r 2 sin 2 = r 3 cos 3 θ r 2 = r cos 3 θ. Now, the distance of (
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This note was uploaded on 10/20/2011 for the course MAC 2311 taught by Professor All during the Fall '08 term at University of Florida.