Does the following multivariable limit exist?

Question

What should be the value of $A$ for $\lim_{(x,y)\to (0,0) } f(x,y) $ to exist?

$$ f(x,y) = \begin{cases} \frac{x^4+y^4}{y(x^2+y^2 ) } , \quad y \neq 0 \\ A , \quad y=0 \end{cases} $$

Thanks in advance!

It seems that $\frac{x^4+y^4}{(x^2+y^2 ) } \to 0$ , but $\frac{1}{y} \to \infty$ does not help me that much. In addition, substituting polar coordinates does not give me anything useful, because of the expression $\cot(\theta) \cos^3 (\theta) $ that is not bounded. Any help will be appreciated

Answer 1

We have that

• for $x=y=t\to 0 \implies \large{\frac{x^4+y^4}{y(x^2+y^2 ) } =\frac{2t^4}{2t^3}=t\to 0}$

• for $x=t$ and $y=t^2$ with $t\to 0 \implies \large{\frac{x^4+y^4}{y(x^2+y^2 ) } =\frac{t^4+t^8}{t^4+t^6}=\frac{1+t^4}{1+t^2}\to 1}$