Question

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

Short answer

Despite the given function value \(f(2,3)=4\), you can't draw any conclusions about the limit of the function \(f(x, y)\) as \((x, y)\) approaches \((2,3)\), because the concept of limit in multivariable calculus necessitates that the function approaches a specific value from all possible paths. Therefore, the limit at a point can differ from the function's value at that point.

Step-by-step solution

Understanding the concept of limit

In a multi-variable calculus, a limit exists at a point if the function approaches a particular value when the point \((x, y)\) reaches a certain point from all possible directions. It means the function should have the same value regardless of the approach path. It's a fundamental concept that guarantees the continuity of a function.

Interpretation of the function value

The function \(f(2,3)\) is given to be 4. This means that the function value at the point \((2,3)\) is 4. This tells us about the state of the function at the specific point, but it doesn't guarantee the function's consistency or limit to approach this value as \((x, y)\) approaches \((2,3)\) from various directions.

Conclusion

Although the function \((x, y)\) equals 4 at \((2,3)\), you cannot conclude that the limit of the function \(f(x, y)\) as \((x, y)\) tends to \((2,3)\) would also be 4. It's because the function's value at a certain point doesn't necessarily have to be the limit at that point, particularly in the context of functions of multiple variables.