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Chapter 13: Problem 19

Evaluate the following limits. $$\lim _{(x, y) \rightarrow(6,2)} \frac{x^{2}-3 x y}{x-3 y}$$

Short Answer

Expert verified
Answer: The limit of the given function as \((x, y) \rightarrow (6, 2)\) is 6.

Step by step solution

01

Substitute the point (6, 2) into the given function

Given the function \(f(x, y) = \frac{x^{2}-3xy}{x-3y}\), we want to find $$\lim _{(x, y) \rightarrow(6,2)} \frac{x^{2}-3 x y}{x-3 y}$$. At first, we try to substitute (6, 2) into the function: $$f(6, 2) = \frac{6^2 - 3(6)(2)}{6 - 3(2)}$$
02

Check for factorization

Before substituting the numbers, let's check if the function can be factorized: $$\frac{x^{2}-3 x y}{x-3 y} = \frac{x(x-3y)}{x-3y}$$ Now we see that there is a common factor of \((x-3y)\) in both the numerator and the denominator of the function.
03

Cancel the common factors

Cancel the common factors of \((x-3y)\) and simplify the function: $$f(x, y) = x$$
04

Substitute the point (6, 2) into the simplified function

Now, we can substitute the point (6, 2) into the simplified function: $$f(6, 2) = 6$$ Thus, the limit of the given function as \((x, y) \rightarrow (6, 2)\) is: $$\lim _{(x, y) \rightarrow(6,2)} \frac{x^{2}-3 x y}{x-3 y} = 6$$

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