limit calculation

The limit is zero.  See below:

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Here's a 3D graph to illustrate:

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Alan, why can you say that theta tends to zero ?

Why can't (x, y) tend to (0, 0) along some direction other than the x-axis ?

(That's not to say that your answer is incorrect).

Better solution:

Let x  =  f*y  where f is a finite constant.

Then ln(1+x*y²)/(x²+y²) becomes  ln(1 + f*y³)/((f²+1)y²)

No matter what the size of f, eventually, as y tends to zero, f*y³ will become much smaller than 1, so ln(1 + f*y³) will tend to f*y³, hence ln(1 + f*y³)/((f²+1)y²)  →  f*y³/((f² + 1)y²)  →  f*y/(f²+1)  →   0

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