Hi there! I'm newly getting introduced to multivariable functions and their limits. I had an assignment of 6 or 7 of them, and I got the majority except for the last. I've tried a few paths of approach and have concluded that the limit is indeed equal to 0. I just cannot prove it.

$\displaystyle \lim_{\{x,y\}\to \{1,0\}} \, \frac{(x-1)^2 \ln (x)}{(x-1)^2+y^2}$

If $\displaystyle 0<\left\|(x-1,y)\right\| = \sqrt{(x-1)^2+y^2}<\delta$,FIXED AFTER EDIT.

then $\displaystyle \left|g(x,y) - 0\right|<\epsilon$

$\displaystyle \left|g(x,y)\right| = \left|\frac{(x-1)^2 \ln (x)}{(x-1)^2+y^2}\right|= \newline \left|\frac{(x-1)^2}{(x-1)^2+y^2}\right|\left|\ln[x]\right| \leq \left|\frac{(x-1)^2}{(x-1)^2}\right| \left|\ln[x]\right| = \left| \ln[x] \right|$

What would be a good choice for $\displaystyle \delta$ so that $\displaystyle \ln[x] < \epsilon$?

But now that I look at it, it seems like I've done the process a tedious way as I'm not employing the IF statement (simply divided it out).

Any advice? Thanks in advance.