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Chapter 12: Problem 27

Use the Two-Path Test to prove that the following limits do not exist. $$\lim _{(x, y) \rightarrow(0,0)} \frac{x+2 y}{x-2 y}$$

Short Answer

Expert verified
Based on the step-by-step analysis, create a short answer question: Question: Using the Two-Path Test, determine if the following limit exists: $$\lim _{(x, y) \rightarrow(0,0)} \frac{x+2 y}{x-2 y}$$ Answer: The limit does not exist.

Step by step solution

01

Path 1: y = 0

Let's consider the path y=0. Along this path, the function becomes: $$\lim _{(x, 0) \rightarrow(0,0)} \frac{x+2(0)}{x-2(0)} = \lim _{x \rightarrow 0} \frac{x}{x}$$ As we approach the origin, this simplifies to: $$\lim _{x \rightarrow 0} 1 = 1$$
02

Path 2: x = y

Now, let's consider the path x=y. Along this path, the function becomes: $$\lim _{(y, y) \rightarrow(0,0)} \frac{y+2 y}{y-2 y} = \lim _{y \rightarrow 0} \frac{3y}{-y}$$ As we approach the origin, this simplifies to: $$\lim _{y \rightarrow 0} -3 = -3$$
03

Conclusion

We've found that along path 1, the limit is 1 and along path 2, the limit is -3. Since the limit for the two paths is different and the limit should be unique, we can conclude that the limit does not exist: $$\lim _{(x, y) \rightarrow(0,0)} \frac{x+2 y}{x-2 y}$$ does not exist.

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