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Chapter 12: Problem 25

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

Short Answer

Expert verified
Question: Find the limit of the following expression as (x, y) approaches (1, 2): $$\lim _{(x, y) \rightarrow(1,2)} \frac{\sqrt{y}-\sqrt{x+1}}{y-x-1}$$ Answer: The limit is equal to $$\frac{\sqrt{2}}{4}$$.

Step by step solution

01

Analyze the function at the given point

At the given point \((1,2)\), the expression under the limit can be observed as follows: $$\frac{\sqrt{y}-\sqrt{x+1}}{y-x-1}$$ Plugging \((x, y) = (1, 2)\) into the expression, we get: $$\frac{\sqrt{2}-\sqrt{1+1}}{2-1-1}$$ This results in a division by zero scenario (denominator becomes zero), so we cannot directly substitute the point into the expression.
02

Simplify the function

To simplify the expression, we will multiply the numerator and the denominator by the conjugate of the numerator. The conjugate of the numerator is given by \((\sqrt{y} + \sqrt{x+1})\). So we have: $$\frac{\sqrt{y}-\sqrt{x+1}}{y-x-1} \cdot \frac{\sqrt{y}+\sqrt{x+1}}{\sqrt{y}+\sqrt{x+1}}$$ Multiplying the numerators and denominators, we get: $$\frac{(\sqrt{y}-\sqrt{x+1})(\sqrt{y}+\sqrt{x+1})}{(y-x-1)(\sqrt{y}+\sqrt{x+1})}$$ Using the difference of squares, this simplifies to: $$\frac{y-(x+1)}{(y-x-1)(\sqrt{y}+\sqrt{x+1})}$$ Now, the expression becomes: $$\frac{y-x-1}{(y-x-1)(\sqrt{y}+\sqrt{x+1})}$$
03

Apply the limit

Now, we simplify the expression to a form where we can apply the limit directly: $$\lim _{(x, y) \rightarrow(1,2)} \frac{y-x-1}{(y-x-1)(\sqrt{y}+\sqrt{x+1})} = \lim _{(x, y) \rightarrow(1,2)} \frac{1}{\sqrt{y}+\sqrt{x+1}}$$ By plugging \((x, y) = (1, 2)\) into the simplified expression, we get: $$\frac{1}{\sqrt{2}+\sqrt{1+1}} = \frac{1}{\sqrt{2}+\sqrt{2}} = \frac{1}{2\sqrt{2}} = \frac{\sqrt{2}}{4}$$ Thus, the result of the limit is: $$\lim _{(x, y) \rightarrow(1,2)} \frac{\sqrt{y}-\sqrt{x+1}}{y-x-1} = \frac{\sqrt{2}}{4}$$

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Most popular questions from this chapter

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