Answer to Question #217526 in Real Analysis for khurram shehzad

Question

 Consider any non-zero point in 𝑅2 and name it (𝑎,𝑏). Then                 

(i). Write any four different paths that passes through your chosen point (𝑎,𝑏).

(ii). Compute the limits of the following function when (𝑥,𝑦) → (𝑎,𝑏) along all these four paths,                            

 𝑓(𝑥,𝑦) = {

(𝑥−𝑎)2(𝑦−𝑏) (𝑥−𝑎)4+(𝑦−𝑏)2

, (𝑥,𝑦) ≠ (𝑎,𝑏) 0, (𝑥,𝑦) = (𝑎,𝑏)  

(iii). Conclude from the results obtained in (ii) and answer whether lim (𝑥,𝑦)→(𝑎,𝑏) 𝑓(𝑥,𝑦) exists or not.  

(iv). Is the function 𝑓(𝑥,𝑦) continuous at the origin? Explain.  

(v). Also calculate 𝑓𝑥(𝑎,𝑏) and 𝑓𝑦(𝑎,𝑏).  

(vi). Write all points of differentiability of 𝑓.

Answer 1

i.

We consider $(a,b) = (0,0)$

Now the four different paths through $(0,0)$ are

$y=x\\ y=x^2\\ y=x^3\\ y=x^4\\$

ii

$f(x,y) = \begin{cases} \frac{(x-a)^2(y-b)}{(x-a)^4(y-)^2} &\text{if } (x,y)\not=(a,b)\\ 0 &\text{if } (x,y)=(a,b) \end{cases}\\\\ But (a,b) = (0,0)\\ f(x,y) = \begin{cases} \frac{x^2y}{x^4y^2} &\text{if } (x,y)\not=(0,0)\\ 0 &\text{if } (x,y)=(0,0) \end{cases}\\\\ Now, \lim\nolimits_{(x,y) \to (0,0)}f(x,y)= \lim\nolimits_{(x,y) \to (0,0)}\frac{x^2y}{x^4y^2}\\ y=x\\ \lim\nolimits_{(x) \to (0)}\frac{x^2*x}{x^4*x^2}=0\\ y=x^2\\ \lim\nolimits_{(x) \to (0)}\frac{x^2*x^2}{x^4*x^4}=0.5\\ y=x^3\\ \lim\nolimits_{(x) \to (0)}\frac{x^2*x^3}{x^4*x^6}=0\\ y=x^4\\ \lim\nolimits_{(x) \to (0)}\frac{x^2*x^4}{x^4*x^8}=0\\$

iii.

We will conclude that along every path chosen the limit of the function exist.

iv.

The function is continuous at the origin along every path chosen except along $y=x^2$ 7

v.

$f(x,y)=\frac{x^2y}{x^4+y^2}\\ f_n= \frac{∂f}{∂x}=\frac{(x^2+y^2)*2xy-x^2y(4x^3)}{(x^4+y^2)^2}\\ f_n(0,0)= Nonexistane\\ f_n= \frac{∂f}{∂y}=\frac{(x^4+y^2)*x^2-x^2y(2y)}{(x^4+y^2)^2}\\ f_n(0,0)= Nonexistane\\$

vi.

The function is differentiable at every point except at (0,0)