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Chapter 12: Problem 12

Evaluate the following limits. $$\lim _{(x, y) \rightarrow(1,-3)}(3 x+4 y-2)$$

Short Answer

Expert verified
Question: Find the limit of the function \(f(x, y) = 3x + 4y - 2\) as \((x, y)\) approaches the point \((1, -3)\). Answer: The limit of the function as \((x, y)\) approaches the point \((1, -3)\) is \(-11\).

Step by step solution

01

Evaluating the two-variable function at the given point

We are given the function \(f(x, y) = 3x + 4y - 2\). Now, we need to evaluate this function at the given point \((1, -3)\). So, let's substitute \(x = 1\) and \(y = -3\) into the function: $$f(1, -3) = 3(1) + 4(-3) - 2$$
02

Simplifying the expression

After substituting the given values into the function, we will simplify the resulting expression: $$f(1, -3) = 3 - 12 - 2$$ $$f(1, -3) = -11$$
03

Finding the limit

Since the function is a continuous polynomial, the limit as \((x, y)\) approaches the point \((1, -3)\) is simply the value of the function at that point. So, the limit is given by: $$\lim _{(x, y) \rightarrow (1, -3)}(3x + 4y - 2) = f(1, -3) = -11$$ So, the limit is: $$\lim _{(x, y) \rightarrow (1, -3)}(3x + 4y - 2) = -11$$

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