I need to prove that
$$\lim \limits_{(x,y)\to(0,0)} \frac{5y^2\cos^2x}{x^4 + y^4}$$
does not exist.
Please tell me why this false positive argument is flawed.
However, if I put this into a delta epsilon limit, I obtain that:
$$\frac{\left|5y^4\cos^2x - (x^4+y^4)L\right|}{\epsilon} < x^4 + y^4 \qquad(1)$$
Then we need to find $\delta$ such that $\sqrt{x^2 + y^2} < \delta$ for any given $\epsilon$. Well because for $x^4 + y^4$ near zero $x^4 + y^4 \le x^2 + y^2$, by the monotonicity of $\sqrt{x}$, $$\sqrt{x^4 + y^4} \le \sqrt{x^2 + y^2}$$ Just use any $$\delta < \sqrt{\frac{\left|5y^4\cos^2x - (x^4+y^4)L\right|}{\epsilon}} \le \sqrt{x^2+y^2}$$
What about this is false?
Is it because there exists $x^4 + y^4 > x^2 + y^2$ so the limit doesn't always hold for any given $\epsilon$?