Answer to Question #335782 in Calculus for Dhanush

Question

Check whether limit of the function of f(x,y)=4x^5y/x^10+3y^2 exists as (x,y) tends to (0,0).

Answer 1

If $x=0$ then $f(0, y)=\dfrac{0(y)}{0+3y^2}=0.$ Therefore

$$f(x, y)\to 0\text{ as} \ (x,y)\to(0,0) \text{ along the }y-\text{axis}$$

For all $x\not=0$

$$f(x, x^5)=\dfrac{4x^5(x^5)}{x^{10}+3(x^5)^2}=1\not=0$$

$$f(x, y)\to 1\not=0\text{ as} \ (x,y)\to(0,0) \text{ along }y=x^5$$

Since we have obtained different limits along different paths, limit

$$\lim\limits_{(x,y)\to(0,0)}f(x, y)=\lim\limits_{(x,y)\to(0,0)}\dfrac{4x^5y}{x^{10}+3y^2}$$

does not exist.