Write down detailed explanations and arguments to justify your calculations. If you wish to use a certain formula, be sure to state it clearly and give a brief justification for it.
Consider the function f(x,y) = (x^4 y)/((x^8)-y(x)^(4)+(y)^2 ) This is an interesting example where the limit as (x, y) Ac†' (0, 0) does not exist, even though the limit along any line and parabola does.
Answer the following questions.
1.) Show that the limit along any line r AcA??~(t) =< at, bt > when t Ac†' 0 exists and equals 0.
2.) Show that the limit along any parabola c AcA??~(t )=< at, bt^2 > when tAc†'0 exists and equals 0
3.) Calculate f(x, y) at the points ((10)^(-1), (10)^(-4)), ((10)^(-5),(10)^(-20)), ((10)^(-20), (10)^(-80)). Do not use a Calculator.
4.) show that lim(x,y) Ac†'(0,0)f(x,y) does not exists. Hint: compute the limit along the curve Y = x^4.