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Chapter 12: Problem 22

Evaluate the following limits. $$\lim _{(x, y) \rightarrow(-1,1)} \frac{2 x^{2}-x y-3 y^{2}}{x+y}$$

Short Answer

Expert verified
Question: Evaluate the following limit or state that it does not exist: $$\lim _{(x, y) \rightarrow(-1,1)} \frac{2 x^{2}-x y-3 y^{2}}{x+y}$$ Answer: The limit does not exist.

Step by step solution

01

State the limit

First, we need to state the given limit problem: $$\lim _{(x, y) \rightarrow(-1,1)} \frac{2 x^{2}-x y-3 y^{2}}{x+y}$$
02

Check if the function is continuous

Before trying different paths, let's verify if the function is continuous at the point by checking its denominator. The denominator is \(x+y\), which is only undefined when x = -y. Therefore, the function is continuous at (-1, 1).
03

Approach along the x-axis

Now we will look at the limit as y remains constant at 1, and x approaches -1: $$\lim_{x \rightarrow -1} \frac{2 x^{2} - x(1) - 3(1)^2}{x + 1}\implies \lim_{x \rightarrow -1} \frac{2 x^{2} - x - 3}{x + 1}$$ To evaluate this limit, we can use factorization: $$\lim_{x \rightarrow -1} \frac{(2x+1)(x-3)}{x + 1}$$ Since we have a removable discontinuity at \(x=-1\), we can cancel it out: $$\lim_{x \rightarrow -1} (2x+1) = (2(-1) + 1) = -1$$
04

Approach along the y-axis

Now we look at the limit as x remains constant at -1, and y approaches 1: $$\lim_{y \rightarrow 1} \frac{2(-1)^2 - (-1)y - 3y^2}{-1 + y}\implies \lim_{y \rightarrow 1} \frac{2 - y - 3y^2}{-1 + y}$$ To evaluate this limit, we use the same technique as in step 3: $$\lim_{y \rightarrow 1} \frac{(1-y)(1-3y)}{-1 + y}$$ Again, we have a removable discontinuity at \(y=1\), so we cancel it out: $$\lim_{y \rightarrow 1} (1-3y) = (1 - 3(1)) = -2$$
05

Compare the results

Now, we compare the results of the two limits we calculated in steps 3 and 4: - Limit when approaching along x-axis: -1 - Limit when approaching along y-axis: -2
06

Reach a conclusion

Since the limit is different when approaching the point (-1, 1) along two different paths (x-axis and y-axis), the limit does not exist: $$\lim _{(x, y) \rightarrow(-1,1)} \frac{2 x^{2}-x y-3 y^{2}}{x+y} \text{ does not exist.}$$

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