Example 4. Find $\lim _{\substack{x \rightarrow 0 \\ y \rightarrow 0}} \frac{x^{2}-y^{2}}{x^{2}+y^{2}}$.
Solution. Dividing the numerator and the denominator by $x^{2}$, we get

$$
\frac{x^{2}-y^{2}}{x^{2}+y^{2}}=\frac{1-\left(\frac{y}{x}\right)^{2}}{1+\left(\frac{y}{x}\right)^{2}}
$$

Since the ratio $\frac{y}{x}$ does not have a limit as $x \rightarrow 0$ and $y \rightarrow 0$ arbitrarily, the given function also does not have a limit.

If we assume that $x \rightarrow 0$ and $y \rightarrow 0$ along the line $y=k x$, then

$$
\lim _{\substack{x \rightarrow 0 \\ y \rightarrow 0}} \frac{x^{2}-y^{2}}{x^{2}+y^{2}}=\lim _{\substack{x \rightarrow 0 \\ y \rightarrow 0}} \frac{1-\left(\frac{y}{x}\right)^{2}}{1+\left(\frac{y}{x}\right)^{2}}=\frac{1-k^{2}}{1+k^{2}}
$$

The limit depends on the coefficient $k$, and for each line $y=k x$ it will be different.