Limits — Question 3

Question 3 – Solution

Strategy. Compare straight lines with the balancing path $y=kx^2$, where $x^4$ and $y^2$ have the same order.

Step 1: Lines Along $y=mx$, $$\frac{x^2(mx)}{x^4+m^2x^2}=\frac{mx}{x^2+m^2}\longrightarrow 0$$ for $m\ne 0$; the path $y=0$ also gives $0$.

Step 2: Parabolas Along $y=kx^2$, $$\frac{x^2(kx^2)}{x^4+k^2x^4}=\frac{k}{1+k^2}.$$ For example, $k=1$ gives $1/2$, whereas $k=0$ gives $0$.

Conclusion. Since two paths produce different values, $$\boxed{\displaystyle \lim_{(x,y)\to(0,0)}\frac{x^2y}{x^4+y^2}\text{ does not exist}.}$$ Agreement along all lines is necessary but not sufficient because nonlinearly approaching paths remain possible.