Q 4. - Calculus

Question

If $\underset{\underset{C_1}{(x,y)\to (-2,0)}}{\lim }g(x,y)=5 and \underset{\underset{C_2}{(x,y)\to (-2,0)}}{\lim }g(x,y)=8$  , where $C_1 and C_2$ are two distinct curves in ${\mathrm{ℝ}}^2$ containing the point $(-2,0)$, what can you say about $\underset{(x,y)\to (-2,0)}{\lim }g(x,y)$?

Step-by-Step Solution

Answer

The limit of the function does not exist.

Step 1: Given information

It is given that $\underset{\underset{C_1}{(x,y)\to (-2,0)}}{\lim }g(x,y)=5 and \underset{\underset{C_2}{(x,y)\to (-2,0)}}{\lim }g(x,y)=8$  here $C_1 and C_2$ are two distinct curves we need to determine what we can say about $\underset{(x,y)\to (-2,0)}{\lim }g(x,y)$.

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.

The function must approach the same value regardless of direction so that we can say the limit exists.

In this case, the limiting values are different so we can say that limit does not exist.