Quiz 4 Solution for MATH 20C, 20091. Evaluate the limit or determine that the limit does not exist.lim(x,y)→(0,0)y2x2+y2Ans.We choose two different path approaching (0,0) and find that there exist two limit valuesalong different paths.First, along they-axis, i.e., whenx= 0,lim(0,y)→(0,0)y2y2=lim(0,y)→(0,0)1 = 1.Secondly, if we choose thex-axis, i.e., wheny= 0,lim(x,0)→(0,0)0x2+ 0=lim(x,0)→(0,0)0 = 0.We checked that along two different coordinate axes, the limit values are different. Hence, the limitdoes not exist.2. Compute the derivative indicated.f(x, y) =xln(y2),fyy(2,3)Ans.Note thatf(x, y) = 2xlny. Then the first and second partial derivatives offwith respecttoyarefy(x, y) =2xy,fyy(x, y) =-2xy2⇒fyy(2,3) =-493. Use linear approximation to
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