How to evaluate $$\lim_{(x,y) \to (0,0)} \frac{ax^2+by^2}{cy^2+dx^2} $$ using $y=mx$?
Here what I've done: \begin{align*} \lim_{(x,y)\to(0,0)}\frac{ax^2 + b(mx)^2}{c(mx)^2 + dx^2}&= \frac{x^2(ax+bm^2)}{x^2(cm^2 + d)}\\ &= \frac{ax + bm^2}{cm^2 +d} \end{align*}
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$\begingroup$You have the right idea but I think some mistakes slipped it:
$$\frac{ax^2+b(mx)^2}{c(mx)^2+dx^2}=\frac{x^2(a+bm^2)}{x^2(cm^2+d)}=\frac{a+bm^2}{cm^2+d}$$
and thus the limit doesn't exist as it depends on the parameter $\;n\;$ .
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