Q 3. - Calculus

Question

If $\underset{\underset{C_1}{(x,y)\to (1,-3)}}{\lim }f(x,y)=5 and \underset{\underset{C_2}{(x,y)\to (1,-3)}}{\lim }f(x,y)=5$ , where $C_1 and C_2$ are two distinct curves in ${\mathrm{ℝ}}^2$ containing the point $(1, -3)$, what can you say about $\underset{(x,y)\to (1,-3)}{\lim }f(x,y)$? 

Step-by-Step Solution

Answer

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

Step 1: Given information

It is given that $\underset{\underset{C_1}{(x,y)\to (1,-3)}}{\lim }f(x,y)=5 and \underset{\underset{C_2}{(x,y)\to (1,-3)}}{\lim }f(x,y)=5$ here $C_1 and C_2$ are two distinct curves we need to determine what we can say about $\underset{(x,y)\to (1,-3)}{\lim }f(x,y)$.

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 the approach is irrelevant when determining the limit unless it passes through the input location.

But the existence of a function at the input point is not compulsory and it is possible that the function exists at $(1,-3)$ and shows different values from the limit at the point $(1,-3)$.

So the value of the function may or may not be equal to 5.