Chapter 12: Q 3. (page 930)

If $\lim_{\underset{C_{1}}{(x, y) \to (1, - 3)}} f (x, y) = 5 a n d \lim_{\underset{C_{2}}{(x, y) \to (1, - 3)}} f (x, y) = 5$ , where $C_{1} a n d C_{2}$ are two distinct curves in $ℝ^{2}$ containing the point $(1, - 3)$ , what can you say about role="math" localid="1652879847340" $\lim_{(x, y) \to (1, - 3)} f (x, y)$ ?

Short Answer

Expert verified

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

Step by step solution

Given information

It is given that $\lim_{\underset{C_{1}}{(x, y) \to (1, - 3)}} f (x, y) = 5 a n d \lim_{\underset{C_{2}}{(x, y) \to (1, - 3)}} f (x, y) = 5$ here $C_{1} a n d C_{2}$ are two distinct curves we need to determine what we can say about $\lim_{(x, y) \to (1, - 3)} f (x, y)$ .

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.

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If lim(x,y)→(1,-3)C1f(x,y)=5andlim(x,y)→(1,-3)C2f(x,y)=5, where C1andC2are two distinct curves in ℝ2containing the point (1,−3), what can you say about role="math" localid="1652879847340" lim(x,y)→(1,-3)f(x,y)?

Short Answer

Step by step solution

Given information

Conclusion from these limits

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