Q. 5 - Calculus

Question

If 

 $\underset{\underset{C}{(x,y)\to (3,-7)}}{\lim }f(x,y)=5$  

for every curve C in $R^2$ containing the point $(3,-7)$,what can you say about $\underset{(\mathrm{x},\mathrm{y})\to (3,-7)}{\lim }f(x,y)?$

What can you say about $f(3,-7)?$

Step-by-Step Solution

Answer

The limit of the function may or may not be 5.

Step 1: Given Information

It is given that $\underset{\underset{C}{(x,y)\to (3,-7)}}{\lim }f(x,y)=5$, where C is a curve in $R^2$containing the point $(3,-7)$

Step 2: Simplifying the limit

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 simple limit can be expressed as $\underset{(\mathrm{x},\mathrm{y})\to (3,-7)}{\lim }f(x,y)$

Step 3: 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. However, the presence of function at the input point or the value of function at that moment are not required. It's extremely likely that the function will become indeterminate or take on a value other than 5 at this point.

hence the value of the function at $f(3,-7)$,may or may not be 5.