Every straight line through the origin gives limit zero: both axes vanish identically, while a nonvertical line has numerator order five and denominator leading order two, leaving positive quotient order three. The nonlinear paths y=c*x^4 instead give the constant c/(1+c^2), which is nonzero for each submitted c. Therefore agreement on every straight line does not establish the multivariable limit, and the limit at the origin does not exist.
RESULT_JSON:{"exponent_p":2,"function":{"numerator_x_power":4,"numerator_y_power":1,"denominator_terms":[{"x_power":8,"y_power":0},{"x_power":0,"y_power":2}]},"origin_value":"0","line_certificate":{"axes_zero":true,"numerator_order":5,"denominator_leading_order":2,"quotient_order":3,"arbitrary_nonzero_slope_limit":"0"},"nonlinear_paths":[{"c":"1","y_x_power":4,"limit":"1/2"},{"c":"2","y_x_power":4,"limit":"2/5"},{"c":"-1/2","y_x_power":4,"limit":"-2/5"}]}
