Question

Find the limit xy/r² where limit x approaches 0.

Answer 1

Answer:

One way to analyze this problem is to change from a rectangular system (x,y) to a polar system (r , θ)  via

x = r cos(θ) ,  y = r sin(θ).    The expression to be analyzed becomes:

 

  [r3 cos3(θ) + r3 sin3(θ) ]/r2  =   r[cos3(θ) + sin3(θ) ]

 

 The expression in the square brackets is   bounded between -1 and +1.  As (0,0) is approached, r approaches zero.

So regardless of what value of θ (direction of approach) is considered, the limit is zero.

Step-by-step explanation:

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