How to evaluate $\lim_{(x,y)\to (0,0)} \frac{x^4+y^4}{2x-y}$?

Question

Find the limit :

$$\lim_{(x,y)\to (0,0)} \frac{x^4+y^4}{2x-y}$$

I have tried using polar substitution but the problem is the denominator. Denominator can go to $0$ for some value of theta and is not dependent on only $r$.

Answer 1

Let $y=2x-t$. Then

$$\frac{x^4+(2x-t)^4}{t}=17x^4/t+P(t,x),$$

where $P(t,x)$ is a polynomial in $t$ and $x$. Now let $t=x^5$. This will cause the limit to blowup. Tracing this back, let $y:=2x-x^5$ and confirm by usual means that it blows up.

Answer 2

Hint: let $y=2x-x^4$. Then the smallest order term in both the numerator and denominator will be quartic, so the limit will be non-zero.

Answer 3

Along the line $y=x$, the limit is zero, but along the line $y=2x$ we always have zero in the denominator, but apart from $(0,0)$, not in the numerator, so the limit does not exist.