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Chapter 12: 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
Question: Evaluate the limit of the two-variable function $$\frac{x^{2}-3 x y}{x-3 y}$$ as \((x,y) \rightarrow (6,2)\). Answer: The limit of the given function when \((x, y) \rightarrow (6,2)\) is equal to \(6\).

Step by step solution

01

Simplify the expression (if possible)

In the given expression, we have: $$\frac{x^{2}-3 x y}{x-3 y}$$. Check if we can factor any terms to simplify this expression. On the numerator, we can notice they have a common term "x", so let's take that out as a factor: \(x(x - 3y)\). The expression thus becomes: $$\frac{x(x-3 y)}{x-3 y}$$. Now, we can see that \((x-3 y)\) term in the numerator and denominator cancels out, leaving us with "x".
02

Substitute the limit values

Now that we have a simplified expression, let's substitute the limit values of \((x,y) \rightarrow (6,2)\) into the function: $$x = 6$$
03

Evaluate the limit

The simplified expression is a constant function, so the limit is the same as the value of the function: $$\lim_{(x, y) \rightarrow (6,2)} \frac{x^{2}-3 x y}{x-3 y} = \lim_{(x,y) \rightarrow (6,2)} x = 6$$ So, the limit of the given function when \((x, y) \rightarrow (6,2)\) is equal to \(6\).

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