Q 6. - Calculus

Question

If $g : R^3\to R$ is continuous at the point $(a,b,c) and C$ is a curve in $R^3$ containing the point $(a,b,c)$,what can we say about $\underset{\underset{C}{(x,y,z)\to (a,b,c)}}{\lim }g(x,y,z)?$

Step-by-Step Solution

Answer

The limit evaluates the output value to which the function approaches as input approaches the given point. 

Step 1: Given Information

The given limits are $\underset{\underset{C}{(x,y,z)\to (a,b,c)}}{\lim }g(x,y,z)$, where C is a curve in   containing the point $(a,b,c)$ 

Step 2: Conclusion from these limits.

The output that the function tends to approach as the input approaches the point is determined by evaluating the function's limit at a point. The path of approach is irrelevant for determining the limit unless it passes via the input location. As a result, regardless of the approach path, the limiting value will remain constant. The function may or may not travel through the point as well. It's conceivable that the function is now uncertain. As the input approaches the designated point, the limit evaluates the output value to which the function approaches.