Multivariable limit $\lim_{(x,y)\to (0,0)} \frac {x^2 + y^2}{\sqrt{x^2 +y^2 + 1} - 1}$

Remember the difference of squares algebraic identity.

$$ A^2 - B^2 = (A - B)(A + B) $$

Why is that useful? With $A = \sqrt{x^2 + y^2 + 1}$ and $B = 1$, the denominator of your expression is $A - B$. With that in mind,

$$ \begin{align} \frac{x^2 + y^2}{\sqrt{x^2 + y^2 + 1} - 1} &= \frac{x^2 + y^2}{\sqrt{x^2 + y^2 + 1} - 1} \cdot \frac{\sqrt{x^2 + y^2 + 1} + 1}{\sqrt{x^2 + y^2 + 1} + 1} \\ &= \frac{(x^2 + y^2)\left(\sqrt{x^2 + y^2 + 1} + 1 \right)}{x^2 + y^2} \\ &= \sqrt{x^2 + y^2 + 1} + 1 \end{align} $$

Now, you can evaluate the limit as $(x, y) \to (0, 0)$ simply by evaluation, since this expression is continuous at the origin.

$$ \lim_{(x, y) \to (0, 0)} \sqrt{x^2 + y^2 + 1} + 1 = \sqrt{0^2 + 0^2 + 1} + 1 = 2. $$

The following "trick" can be useful. Let $x=r\cos\theta$, $y=r\sin\theta$. Then we are finding $$\lim_{r\to 0}\frac{r^2}{\sqrt{r^2+1}-1}.$$ This can be computed using any of the usual one variable techniques.