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Chapter 12: Problem 23

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

Short Answer

Expert verified
Answer: The limit of the function as (x, y) approaches (2, 2) is 2.

Step by step solution

01

Substitute the limit values into the function

First, try substituting the limit values (2, 2) directly into the function: $$\frac{(2)^{2}-4}{2(2)-2(2)}$$ This results in the indeterminate form 0/0, so we need to simplify the function further.
02

Factor the numerator and denominator

Factor the expressions in the numerator and denominator: $$\frac{y^2 - 4}{xy - 2x} = \frac{(y-2)(y+2)}{x(y-2)}$$
03

Cancel the common factor

Now, we can cancel the common factor (y-2) from the numerator and denominator: $$\frac{(y-2)(y+2)}{x(y-2)} = \frac{y+2}{x}$$
04

Substitute the limit values again

With the simplified function, substitute the limit values (2, 2) into the function again: $$\frac{2+2}{2} = \frac{4}{2}$$
05

Evaluate the limit

Now we can evaluate the limit: $$\lim_{(x, y) \rightarrow (2,2)} \frac{y^2 - 4}{xy - 2x} = \frac{4}{2} = 2$$ So the limit of the given function as (x, y) approaches (2, 2) is 2.

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