Use the two-path test to prove that the following limit does not exist. lim_(x,

Question

Use the two-path test to prove that the following limit does not exist. lim_(x, y) rightarrow (0, 0) y^4 - 2x^2/y^4 + x^2 The body mass index (BMI) for an adult human is given by the function B = w/h^2 where w is the weight measured in kilograms and h is the height measured in meters. (The BMI for units of pounds and inches is B = 703 w/h62.) Find the rate of change of the BMI with respect to weight at a constant height For Fixed h, is the BMI an increasing or decreasing function of w? Find the rate of change of the BMI with respect to height at a constant weight. For fixed w, is the BMI an increasing or decreasing function of h?

Explanation / Answer

1)lim(x,y)->(0,0) (y4-2x2)/(y4+x2)

along the path y =0

=lim(x)->(0) (04-2x2)/(04+x2)

=lim(x)->(0) (-2x2)/(x2)

=lim(x)->(0) -2

=-2

lim(x,y)->(0,0) (y4-2x2)/(y4+x2)

along the path x =0

=lim(y)->(0) (y4-2*02)/(y4+02)

=lim(y)->(0) y4/y4

=lim(y)->(0) (1)

=1

different paths give different limits. so limit doesnot exist.

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