Find the following limit or prove that it does not exist Let

Find the following limit or prove that it does not exist: Let f(x, y, z) = . Find the directions of maximum and minimum change of f at the point (1, 2, -2), and the values of the maximum and minimum rates of change

Solution

a)

limit doesn\'t exists because when we take the path x tends to 0 then y tends to 0 then we\'ll get 0*(cosy-1)/y^3

=0

but when we take path x=y then y tends to 0 then we got (cosy-1)/(2*y^2) then y tends to 0 which is equal to -1/4

so we got different limits when going through different path so limit doesn\'t exist.

b)

now to find change we have to calculate f_x(1,2,-2), f_y(1,2,-2), f_z(1,2,-2)

f_x(1,2,-2) = 2*x/((x²+y²+z²)) = 2/5

f_y(1,2,-2) = 2*y/((x²+y²+z²)) = 4/5

f_z(1,2,-2) = 2*z/((x²+y²+z²)) = -4/5

so minimum chnage will be in x direction and maximum change will be in y and z direction but in different sense