1. The problem is to understand the concept of limits in multivariable calculus. 2. The limit of a function $f(x,y)$ as $(x,y)$ approaches a point $(a,b)$ is defined as $$\lim_{(x,y) \to (a,b)} f(x,y) = L$$ if for every number $\epsilon > 0$, there exists a $\delta > 0$ such that whenever $\sqrt{(x-a)^2 + (y-b)^2} < \delta$, it follows that $|f(x,y) - L| < \epsilon$. 3. Important rules: - The limit must be the same regardless of the path taken to approach $(a,b)$. - If the limit depends on the path, the limit does not exist. 4. To evaluate limits, try substituting the point directly if the function is continuous there. 5. If direct substitution leads to an indeterminate form, try approaching along different paths (e.g., $y=mx$, $y=x^2$) to check if the limit is consistent. 6. Example: Evaluate $$\lim_{(x,y) \to (0,0)} \frac{xy}{x^2 + y^2}$$. 7. Approach along $y=mx$: $$\lim_{x \to 0} \frac{x(mx)}{x^2 + (mx)^2} = \lim_{x \to 0} \frac{m x^2}{x^2 + m^2 x^2} = \lim_{x \to 0} \frac{m}{1 + m^2} = \frac{m}{1 + m^2}$$ 8. Since the limit depends on $m$, the slope of the path, the limit does not exist. Final answer: The limit does not exist because it depends on the path taken.