Question

If \(\lim _{(x, y) \rightarrow(2,3)} f(x, y)=4,\) can you conclude anything about \(f(2,3) ?\) Give reasons for your answer.

Short answer

No, one can't conclude anything about \(f(2,3)\) from \(\lim _{(x, y) \rightarrow(2,3)} f(x, y)=4\). The limit does not guarantee the function's value at that specific point.

Step-by-step solution

Understanding the Problem

The unknown here is whether the function \(f\) at the point \((2, 3)\) equals the limit \(\lim _{(x, y) \rightarrow(2,3)} f(x, y)\). This involves understanding how limits in multivariable functions work and how they relate to the function values at specific points.

Analyzing the Limit of the Function

\(\lim _{(x, y) \rightarrow(2,3)} f(x, y)=4\) suggests that as \(x\) approaches 2 and \(y\) approaches 3, the function \(f(x, y)\) gets arbitrarily close to 4. This, however, gives no information about the value of the function \(f\) exactly at the point \((2, 3)\).

Coming to a Conclusion

No, nothing can be concluded about \(f(2,3)\) based solely on \(\lim _{(x, y) \rightarrow(2,3)} f(x, y)=4\). They might be equal, but they are not necessarily so. Explanation: The limit of a function at a certain point refers to the value that the function approaches as its variable approaches that point. It does not always mean that the function actually reaches that value at that exact point, hence nothing can be definitively stated about the value of \(f\) at \((2, 3)\).