TITLE: Multivariable Limit
QUESTION [4 upvotes]: I am trying to  find the following limit:
$$\lim_{(x,y) \to (0,0)} \frac{\ln(e+3x^2+3y^2+x^2y^3+x^3y^2)-\sqrt{1+2x^2+2y^2+x^4+y^4}}{\ln(1+x^2+y^2+x^4+y^4)}$$
I solved the limit along $x=0$ , $y=0$ and $x=y$ and I got $\frac{3}{e} -1$ for all of them. I then used wolfarm to check whether the limit exists or not , it exist and equals to the value mentioned above, but I'm not able to find a way to show that along all the paths to $(0,0)$ the limit equals to $\frac{3}{e} -1$

REPLY [2 votes]: Ignoring terms of degree $>2$, and using polar coordinates your limit becomes
$$\lim_{\rho \to 0} \frac{\ln (e+3\rho^2) - \sqrt{1+ 2 \rho^2}}{\ln(1+ \rho^2)}$$
which is easily computed (call $r= \rho^2$ to get further simplification), and equals $\frac{3}{e}-1$
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