Chapter 12: Q 4. (page 930)

If $\lim_{\underset{C_{1}}{(x, y) \to (- 2, 0)}} g (x, y) = 5 a n d \lim_{\underset{C_{2}}{(x, y) \to (- 2, 0)}} g (x, y) = 8$ , where $C_{1} a n d C_{2}$ are two distinct curves in $ℝ^{2}$ containing the point $(- 2, 0)$ , what can you say about $\lim_{(x, y) \to (- 2, 0)} g (x, y)$ ?

Short Answer

Expert verified

The limit of the function does not exist.

Step by step solution

Given information

It is given that $\lim_{\underset{C_{1}}{(x, y) \to (- 2, 0)}} g (x, y) = 5 a n d \lim_{\underset{C_{2}}{(x, y) \to (- 2, 0)}} g (x, y) = 8$ here $C_{1} a n d C_{2}$ are two distinct curves we need to determine what we can say about $\lim_{(x, y) \to (- 2, 0)} g (x, y)$ .

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.

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.

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If $\lim_{\underset{C_{1}}{(x, y) \to (- 2, 0)}} g (x, y) = 5 a n d \lim_{\underset{C_{2}}{(x, y) \to (- 2, 0)}} g (x, y) = 8$ , where $C_{1} a n d C_{2}$ are two distinct curves in $ℝ^{2}$ containing the point $(- 2, 0)$ , what can you say about $\lim_{(x, y) \to (- 2, 0)} g (x, y)$ ?

Short Answer

Step by step solution

Given information

Conclusion from these limits.

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If lim(x,y)→(-2,0)C1g(x,y)=5andlim(x,y)→(-2,0)C2g(x,y)=8 , where C1andC2are two distinct curves in ℝ2containing the point (-2,0), what can you say about lim(x,y)→(-2,0)g(x,y)?

Short Answer

Step by step solution

Given information

Conclusion from these limits.

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Most popular questions from this chapter

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If $\lim_{\underset{C_{1}}{(x, y) \to (- 2, 0)}} g (x, y) = 5 a n d \lim_{\underset{C_{2}}{(x, y) \to (- 2, 0)}} g (x, y) = 8$ , where $C_{1} a n d C_{2}$ are two distinct curves in $ℝ^{2}$ containing the point $(- 2, 0)$ , what can you say about $\lim_{(x, y) \to (- 2, 0)} g (x, y)$ ?