1. The problem is to explore the expression $0^0$ using limits and understand if it can be represented as 0. 2. The expression $0^0$ is an indeterminate form in mathematics, meaning its value depends on the limiting process. 3. To analyze $0^0$, consider the limit of $f(x,y) = x^y$ as $(x,y) \to (0,0)$. 4. For example, take the limit $\lim_{x \to 0^+} x^x$. 5. Rewrite $x^x$ using exponentials: $x^x = e^{x \ln x}$. 6. As $x \to 0^+$, $\ln x \to -\infty$ but $x \to 0$, so $x \ln x \to 0$. 7. Therefore, $\lim_{x \to 0^+} x^x = e^0 = 1$. 8. Alternatively, consider $\lim_{y \to 0^+} 0^y = 0$ since any positive power of 0 is 0. 9. These different limits show $0^0$ can approach different values depending on the path. 10. Hence, $0^0$ is indeterminate and cannot be universally defined as 0 using limits. Final answer: $0^0$ is an indeterminate form and cannot be represented as 0 by limits.